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Model PCEX

Source: vignettes/Model_PCEX.Rmd
Model_PCEX.Rmd

This guide explains how to use the godley package to create PCEX models — models with government money and portfolio choice with expectations, as described by Wynne Godley and Marc Lavoie in Chapter 4 of Monetary Economics. An Integrated Approach to Credit, Money, Income, Production and Wealth.

Model PCEX - random shocks

In this example, expectations follow a random process, modeled by an equation with a new variable “Ra,” which is normally distributed with a mean of 0:
Yd_e = Yd * (1 + Ra).

To create this model, first initialize an empty SFC model:

# Create empty model
model_pcex <- create_model(name = "SFC PCEX")
#> Empty model created

Then, add variables to the model:

# Add variables
model_pcex <- model_pcex |>
  add_variable("B_cb", desc = "") |>
  add_variable("H_s", desc = "") |>
  add_variable("B_s", desc = "") |>
  add_variable("B_h", desc = "") |>
  add_variable("H_d1", desc = "") |>
  add_variable("H_d", desc = "") |>
  add_variable("H_h", desc = "") |>
  add_variable("C", desc = "") |>
  add_variable("V", desc = "") |>
  add_variable("T_x", desc = "") |>
  add_variable("Y", desc = "Income = GDP") |>
  add_variable("Yd", desc = "Disposable income of households") |>
  add_variable("Yd_e") |>
  add_variable("B_d") |>
  add_variable("V_e") |>
  add_variable("Ra") |>
  add_variable("alpha1", init = 0.6, desc = "Propensity to consume out of income") |>
  add_variable("alpha2", init = 0.4, desc = "Propensity to consume out of wealth") |>
  add_variable("theta", init = 0.2, desc = "Tax rate") |>
  add_variable("r", init = 0.025, desc = "") |>
  add_variable("G", init = 20, desc = "Government demand") |>
  add_variable("lambda0", init = 0.635, desc = "") |>
  add_variable("lambda1", init = 0.05, desc = "") |>
  add_variable("lambda2", init = 0.01, desc = "")

Define the relationships between variables by adding equations:

# Add equations
model_pcex <- model_pcex |>
  add_equation("Y = C + G", desc = "") |>
  add_equation("Yd = Y - T_x + r[-1] * B_h[-1]") |>
  add_equation("T_x = theta * (Y + r[-1] * B_h[-1])") |>
  add_equation("V = V[-1] + (Yd - C)") |>
  add_equation("C = alpha1 * Yd_e + alpha2 * V[-1]") |>
  add_equation("B_d =  V_e * lambda0 + V_e * lambda1 * r - lambda2 * Yd_e") |>
  add_equation("H_d1 = V_e * (1 - lambda0) - V_e * lambda1 * r + lambda2 * Yd_e") |>
  add_equation("H_d = V_e - B_d") |>
  add_equation("V_e = V[-1] + (Yd_e - C)") |>
  add_equation("H_h = V - B_h") |>
  add_equation("B_h = B_d") |>
  add_equation("B_s = B_s[-1] + (G + r[-1] * B_s[-1]) - (T_x + r[-1] * B_cb[-1])") |>
  add_equation("H_s = H_s[-1] + B_cb - B_cb[-1]") |>
  add_equation("B_cb = B_s - B_h") |>
  add_equation("Yd_e = Yd * (1 + Ra)") |>
  add_equation("Ra = rnorm(1, 0, 0.05)") |>
  add_equation("H_h = H_s", hidden = T)

Now, you can simulate the model (in this example, the baseline scenario over 100 periods using the Newton method):

# Simulate model
model_pcex <- simulate_scenario(model_pcex,
  scenario = "baseline", max_iter = 350, periods = 100,
  hidden_tol = 0.1, tol = 1e-08, method = "Newton"
)
#> Model prepared successfully
#> Simulating scenario baseline (1 of 1)
#> Scenario(s) successfully simulated

With the simulation estimated, you can create a plot to visualize the results for the variables of interest:

# Plot results
plot_simulation(
  model = model_pcex, scenario = c("baseline"), from = 1, to = 50,
  expressions = c("H_h", "H_d")
)

Note: The above example uses the new pipe operator (|>), which requires R 4.1 or later.

Model PCEX1 - adaptive expectations

In this scenario, we assume that expectations about current disposable income are based on the actual disposable income from the previous period.

To apply this model, first create an empty SFC model:

# Create empty model
model_pcex1 <- create_model(name = "SFC PCEX1")
#> Empty model created

Next, define relevant variables:

# Add variables
model_pcex1 <- model_pcex1 |>
  add_variable("B_cb", desc = "") |>
  add_variable("H_s", desc = "") |>
  add_variable("B_s", desc = "") |>
  add_variable("B_h", desc = "") |>
  add_variable("H_d1", desc = "") |>
  add_variable("H_d", desc = "") |>
  add_variable("H_h", desc = "") |>
  add_variable("C", desc = "") |>
  add_variable("V", desc = "") |>
  add_variable("T_x", desc = "") |>
  add_variable("Y", desc = "Income = GDP") |>
  add_variable("Yd", desc = "Disposable income of households") |>
  add_variable("Yd_e") |>
  add_variable("B_d") |>
  add_variable("V_e") |>
  add_variable("alpha1", init = 0.6, desc = "Propensity to consume out of income") |>
  add_variable("alpha2", init = 0.4, desc = "Propensity to consume out of wealth") |>
  add_variable("theta", init = 0.2, desc = "Tax rate") |>
  add_variable("r", init = 0.025, desc = "") |>
  add_variable("G", init = 20, desc = "Government demand") |>
  add_variable("lambda0", init = 0.635, desc = "") |>
  add_variable("lambda1", init = 0.05, desc = "") |>
  add_variable("lambda2", init = 0.01, desc = "")

Following that, update the equation for expected disposable income from the previous example to:
Yd_e = Yd[-1]

# Add equations
model_pcex1 <- model_pcex1 |>
  add_equation("Y = C + G", desc = "") |>
  add_equation("Yd = Y - T_x + r[-1] * B_h[-1]") |>
  add_equation("T_x = theta * (Y + r[-1] * B_h[-1])") |>
  add_equation("V = V[-1] + (Yd - C)") |>
  add_equation("C = alpha1 * Yd_e + alpha2 * V[-1]") |>
  add_equation("B_d =  V_e * lambda0 + V_e * lambda1 * r - lambda2 * Yd_e") |>
  add_equation("H_d1 = V_e * (1 - lambda0) - V_e * lambda1 * r + lambda2 * Yd_e") |>
  add_equation("H_d = V_e - B_d") |>
  add_equation("V_e = V[-1] + (Yd_e - C)") |>
  add_equation("H_h = V - B_h") |>
  add_equation("B_h = B_d") |>
  add_equation("B_s = B_s[-1] + (G + r[-1] * B_s[-1]) - (T_x + r[-1] * B_cb[-1])") |>
  add_equation("H_s = H_s[-1] + B_cb - B_cb[-1]") |>
  add_equation("B_cb = B_s - B_h") |>
  add_equation("Yd_e = Yd[-1]") |>
  add_equation("H_h = H_s", hidden = T)

Now, you can simulate the model (in this example, the baseline scenario over 100 periods using the Newton method):

# Simulate model
model_pcex1 <- simulate_scenario(model_pcex1,
  scenario = "baseline", max_iter = 350,
  periods = 100, hidden_tol = 0.1, tol = 1e-08, method = "Newton"
)
#> Model prepared successfully
#> Simulating scenario baseline (1 of 1)
#> Scenario(s) successfully simulated

With the simulation estimated, you can create a plot to visualize the results for the variables of interest:

# Plot results
plot_simulation(
  model = model_pcex1, scenario = c("baseline"), from = 1, to = 50,
  expressions = c("Y")
)

Note: The above example uses the new pipe operator (|>), which requires R 4.1 or later.

Shock - increase in the propensity to consume out of disposable income

Now you can simulate the impact on the economy if the propensity to consume out of disposable income increases to 0.7.

First, create an empty shock object and add a new equation for alpha1:

# Create empty shock
shock_pcex1 <- create_shock()
#> Shock object created

# Add shock equation with increased government expenditures
shock_pcex1 <- add_shock(shock_pcex1,
  variable = "alpha1",
  value = 0.7,
  desc = "Increase in the propensity to consume out of disposable income",
  start = 5, end = 50
)

# Create new scenario with this shock
model_pcex1 <- add_scenario(model_pcex1,
  name = "expansion", origin = "baseline",
  origin_start = 1,
  origin_end = 100,
  shock = shock_pcex1
)

Now you can simulatem the shock scenario:

# Simulate shock
model_pcex1 <- simulate_scenario(model_pcex1,
  scenario = "expansion", max_iter = 350, periods = 100,
  hidden_tol = 0.1, tol = 1e-08, method = "Newton"
)
#> Simulating scenario expansion (1 of 1)
#> Scenario(s) successfully simulated

Finally, once you have the results, you can plot national income, the evolution of consumption, expected disposable income, and lagged wealth.

# Plot results
plot_simulation(
  model = model_pcex1, scenario = c("expansion"), from = 1, to = 50,
  expressions = c("Y")
)
# Plot results
plot_simulation(
  model = model_pcex1, scenario = c("expansion"), from = 1, to = 50,
  expressions = c("Yd_e", "C", "V")
)

Model PCEX2 - where the propensity to consume reacts negatively to higher interest rates

Here, we assume that expectations regarding current disposable income are based on the previous period’s actual disposable income, as in the previous example. However, the propensity to consume now reacts negatively to higher interest rates (decreases as interest rates rise). To apply this model, we need to add two new variables, iota and alpha10, to those used in PCEX1, and include an additional equation:
alpha1 = alpha10 - iota * r[-1].

To begin, create an empty SFC model:

# Create empty model
model_pcex2 <- create_model(name = "SFC PCEX2")
#> Empty model created

Then, define the variables for the model:

# Add variables
model_pcex2 <- model_pcex2 |>
  add_variable("B_cb", desc = "") |>
  add_variable("H_s", desc = "") |>
  add_variable("B_s", desc = "") |>
  add_variable("B_h", desc = "") |>
  add_variable("H_d1", desc = "") |>
  add_variable("H_d", desc = "") |>
  add_variable("H_h", desc = "") |>
  add_variable("C", desc = "") |>
  add_variable("V", desc = "") |>
  add_variable("T_x", desc = "") |>
  add_variable("Y", desc = "Income = GDP") |>
  add_variable("Yd", desc = "Disposable income of households") |>
  add_variable("Yd_e") |>
  add_variable("B_d") |>
  add_variable("V_e") |>
  add_variable("iota", init = 4) |>
  add_variable("alpha10", init = 0.7) |>
  add_variable("alpha1", init = 0.6, desc = "Propensity to consume out of income") |>
  add_variable("alpha2", init = 0.4, desc = "Propensity to consume out of wealth") |>
  add_variable("theta", init = 0.2, desc = "Tax rate") |>
  add_variable("r", init = 0.025, desc = "") |>
  add_variable("G", init = 20, desc = "Government demand") |>
  add_variable("lambda0", init = 0.635, desc = "") |>
  add_variable("lambda1", init = 0.05, desc = "") |>
  add_variable("lambda2", init = 0.01, desc = "")

Specify the relationships between variables by adding equations:

# Add equations
model_pcex2 <- model_pcex2 |>
  add_equation("Y = C + G", desc = "") |>
  add_equation("Yd = Y - T_x + r[-1] * B_h[-1]") |>
  add_equation("T_x = theta * (Y + r[-1] * B_h[-1])") |>
  add_equation("V = V[-1] + (Yd - C)") |>
  add_equation("C = alpha1 * Yd_e + alpha2 * V[-1]") |>
  add_equation("B_d =  V_e * lambda0 + V_e * lambda1 * r - lambda2 * Yd_e") |>
  add_equation("H_d1 = V_e * (1 - lambda0) - V_e * lambda1 * r + lambda2 * Yd_e") |>
  add_equation("H_d = V_e - B_d") |>
  add_equation("V_e = V[-1] + (Yd_e - C)") |>
  add_equation("H_h = V - B_h") |>
  add_equation("B_h = B_d") |>
  add_equation("B_s = B_s[-1] + (G + r[-1] * B_s[-1]) - (T_x + r[-1] * B_cb[-1])") |>
  add_equation("H_s = H_s[-1] + B_cb - B_cb[-1]") |>
  add_equation("B_cb = B_s - B_h") |>
  add_equation("Yd_e = Yd[-1]") |>
  add_equation("alpha1 = alpha10 - iota * r[-1]") |>
  add_equation("H_h = H_s", hidden = T)

Now, you can simulate the model (in this example, the baseline scenario over 100 periods using the Newton method):

# Simulate model
model_pcex2 <- simulate_scenario(model_pcex2,
  scenario = "baseline", max_iter = 350,
  periods = 100, hidden_tol = 0.1, tol = 1e-08, method = "Newton"
)
#> Model prepared successfully
#> Simulating scenario baseline (1 of 1)
#> Scenario(s) successfully simulated

With the simulation estimated, you can create a plot to visualize the outcomes for the variables of interest:

# Plot results
plot_simulation(
  model = model_pcex2, scenario = c("baseline"), from = 1, to = 50,
  expressions = c("Y")
)

Note: The above example uses the new pipe operator (|>), which requires R 4.1 or later.

Shock - increase in the rate of interest on bills

Now, you can simulate the effects on the economy of a 100-point increase in the interest rate on bills.

To start, create an empty shock object and add a new equation for r:

# Create empty shock
shock_pcex2 <- create_shock()
#> Shock object created

# Add shock equation with increased government expenditures
shock_pcex2 <- add_shock(shock_pcex2,
  variable = "r",
  value = 0.035,
  desc = "Increase in the rate of interest on bills",
  start = 5, end = 50
)

# Create new scenario with this shock
model_pcex2 <- add_scenario(model_pcex2,
  name = "expansion", origin = "baseline",
  origin_start = 1,
  origin_end = 100,
  shock = shock_pcex2
)

Afterward, you can simulate the shock scenario:

# Simulate shock
model_pcex2 <- simulate_scenario(model_pcex2,
  scenario = "expansion", max_iter = 350, periods = 100,
  hidden_tol = 0.1, tol = 1e-08, method = "Newton"
)
#> Simulating scenario expansion (1 of 1)
#> Scenario(s) successfully simulated

With these results, you can visualize national income, the evolution of consumption, expected disposable income, and lagged wealth:

# Plot results
plot_simulation(
  model = model_pcex2, scenario = c("expansion"), from = 1, to = 50,
  expressions = c("Y")
)
# Plot results
plot_simulation(
  model = model_pcex2, scenario = c("expansion"),
  from = 1, to = 50,
  expressions = c("Yd", "C", "V")
)

References

For more details on PC models with expectations (PCEX), see Chapter 4 of Monetary Economics: An Integrated Approach to Credit, Money, Income, Production, and Wealth.