Behavioral Equations GL06PCEX#

Initialization Equations#

Initialize the behavior of the Godley-Lavoie 2006 PCEX model. Within the book the initialization is generally to set all non-scenario variables to zero. Accordingly

\begin{align} C(0) &= 0 \\ G(0) &= 0 \\ Y(0) &= 0 \\ T(0) &= 0 \\ YD(0) &= 0 \\ V(0) &= 0 \\ H_s(0) &= 0 \\ H_h(0) &= 0 \\ B_h(0) &= 0 \\ B_s(0) &= 0 \\ B_{CB}(0) &= 0 \\ r(0) &= 0 \\ \end{align}

Step Equations#

  1. Central Bank Bill Holdings

Calculate the central bank bill holdings.

\begin{align} B_{CB}(t) = B_{s}(t) - B_{h}(t) \end{align}

  1. Central Bank Money Stock

Calculate the central bank money stock.

\begin{align} H_{s}(t) = H_{s}(t-1) + (B_{CB}(t) - B_{CB}(t-1)) \end{align}

  1. Central Bank Profits

Calculate the central bank profits (income on bills held).

\begin{align} r(t-1)B_{CB}(t-1) \end{align}

  1. Consumption

Calculate the consumption.

\begin{align} C(t) = \alpha_1 YD^e(t) + \alpha_2 V(t-1) \end{align}

  1. Disposable Income

Calculate the disposable income.

\begin{align} YD(t) = Y(t) - T(t) + r(t-1)B_h(t-1) \end{align}

  1. Expected Disposable Income

The expected disposable income is simply the prior period’s disposable income. Equation (3.20) in the book.

\begin{align} YD^e(t) = YD(t-1) \end{align}

  1. Expected Wealth

Calculate the expected wealth.

\begin{align} V^e(t) = V(t-1) + YD^e(t) - C(t) \end{align}

  1. Government Bill Issuance

Calculate the government bill issuance.

\begin{align} B_s(t) = B_s(t-1) + (G(t) - r(t-1)B_s(t-1)) - (T(t) + r(t-1)B_{CB}(t-1)) \end{align}

  1. Household Bill Demand

Calculate the household bill demand.

\begin{align} \frac{B_h(t)}{V^e(t)} = \lambda_0 + \lambda_1 r(t) - \lambda_2 \frac{YD^e(t)}{V^e(t)} \end{align}

  1. Household Bill Holdings

Calculate the household bill holdings.

\begin{align} B_h(t) = B_h(t-1) + (B_h^d(t) - B_h(t-1)) \end{align}

  1. Household Money Stock

Calculate the household deposits as a residual.

\begin{align} H_h(t) = V(t) - B_h(t) \end{align}

  1. Interest Earned On Bills Household

Calculate the interest earned on bills by the household.

\begin{align} r(t-1)B_h(t-1) \end{align}

  1. National Income

Calculate the national income based on the closed-form solution derived in the documentation. The closed-form solution is used to avoid the need to solve the system of equations iteratively, thus preserving the differentiability of the model trajectory.

\begin{align} Y(t) = C(t) + G(t) \end{align}

  1. Taxes

Calculate the taxes.

\begin{align} T(t) = \theta (Y(t) + r(t-1)B_h(t-1)) \end{align}

  1. Wealth

Calculate the wealth.

\begin{align} V(t) = V(t-1) + YD(t) - C(t) \end{align}