Behavioral Equations GL06LP2#

Initialization Equations#

Initialize the behavior of the Godley-Lavoie 2006 LP2 model. Extends LP initialization by adding TargetProportion and setting the initial bond price from the scenario.

\begin{align} TP(0) = 0 \\ p_{bl}(0) = p_{bl}^{\text{init}} \end{align}

Step Equations#

  1. Bond Coupon Income Household

Calculate the coupon income from bonds held by the household. Each bond pays a coupon of 1 monetary unit per period, so bond coupon income is simply the number of bonds held last period.

\begin{align} BL_h(t-1) \end{align}

  1. Bond Yield

Calculate the bond yield (coupon / price). Since the coupon is 1 per bond, the yield is simply the inverse of the bond price.

\begin{align} r_{bl}(t) = \frac{1}{p_{bl}(t)} \end{align}

  1. Capital Gains

Calculate the capital gains on bonds. Capital gains arise from changes in the bond price applied to the stock of bonds held at the beginning of the period.

\begin{align} CG(t) = \left(p_{bl}(t) - p_{bl}(t-1)\right) \cdot BL_h(t-1) \end{align}

  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_b(t-1) \cdot B_{CB}(t-1) \end{align}

  1. Compute Bond Price

Compute the endogenous bond price via the target-proportion mechanism. When the target proportion exceeds the upper bound, the Treasury lets bond prices drift upwards. When below the lower bound, prices drift downwards.

\begin{align} z_1 = 1 \text{ iff } TP > \text{top} z_2 = 1 \text{ iff } TP < \text{bot} p_{bl}(t) = (1 + z_1 \cdot \beta - z_2 \cdot \beta) \cdot p_{bl}(t-1) + \text{add1} \end{align}

  1. Consumption

Calculate consumption.

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

  1. Disposable Income

Calculate the disposable income. Disposable income includes national income minus taxes plus interest on bills and bond coupon income.

\begin{align} YD_r(t) = Y(t) - T(t) + r_b(t-1) \cdot B_h(t-1) + BL_h(t-1) \end{align}

  1. Expected Bond Price

Calculate the expected bond price using adaptive expectations. The expected bond price adjusts towards the actual bond price with an error-correction mechanism.

\begin{align} p_{bl}^e(t) = p_{bl}^e(t-1) - \beta_e \cdot (p_{bl}^e(t-1) - p_{bl}(t)) + \text{add} \end{align}

  1. Expected Capital Gains

Calculate the expected capital gains on bonds. In the base LP model with static expectations (pebl = pbl), expected capital gains are zero.

\begin{align} CG^e(t) = \chi \cdot (p_{bl}^e(t) - p_{bl}(t)) \cdot BL_h(t) \end{align}

  1. Expected Disposable Income

Calculate the expected disposable income. Expectations are adaptive: expected disposable income equals the prior period’s actual disposable income.

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

  1. Expected Return On Bonds

Calculate the expected return on bonds. The expected return combines the current yield with the expected capital gain from bond price changes.

\begin{align} ERr_{bl}(t) = r_{bl}(t) + \chi \cdot \frac{p_{bl}^e(t) - p_{bl}(t)}{p_{bl}(t)} \end{align}

  1. Expected Wealth

Calculate the expected wealth. Expected wealth uses expected disposable income instead of actual disposable income, but actual capital gains (which are known at the start of the period since bond prices are exogenous).

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

  1. Government Bill Issuance

Calculate the government bill issuance. The government budget constraint determines the supply of bills: new bills finance the deficit net of bond issuance revenue.

\begin{align} B_s(t) = B_s(t-1) + G(t) + r_b(t-1) \cdot B_s(t-1) + BL_s(t-1) - T(t) - r_b(t-1) \cdot B_{CB}(t-1) - (BL_s(t) - BL_s(t-1)) \cdot p_{bl}(t) \end{align}

  1. Government Bond Supply

Calculate the government bond supply. Bond supply equals bond demand from households.

\begin{align} BL_s(t) = BL_h(t) \end{align}

  1. Household Bill Demand

Calculate the household bill demand using Tobin portfolio choice.

\begin{align} B_d(t) = V^e(t) \cdot \lambda_{20} + V^e(t) \cdot (\lambda_{22} \cdot r_b(t) + \lambda_{23} \cdot ERr_{bl}(t)) + \lambda_{24} \cdot YD_r^e(t) \end{align}

  1. Household Bill Holdings

Calculate the household bill holdings.

\begin{align} B_h(t) = B_d(t) \end{align}

  1. Household Bond Demand

Calculate the household bond demand using Tobin portfolio choice. The demand is in number of bonds (value / price).

\begin{align} BL_d(t) = \frac{V^e(t) \cdot \left(\lambda_{30} + \lambda_{32} \cdot r_b(t) + \lambda_{33} \cdot ERr_{bl}(t) + \lambda_{34} \cdot \frac{YD_r^e(t)}{V^e(t)}\right)} {p_{bl}(t)} \end{align}

  1. Household Bond Holdings

Calculate the household bond holdings.

\begin{align} BL_h(t) = BL_d(t) \end{align}

  1. Household Cash Demand

Calculate the household cash demand as a residual.

\begin{align} H_d(t) = V^e(t) - B_d(t) - p_{bl}(t) \cdot BL_d(t) \end{align}

  1. Household Cash Stock

Calculate the household cash stock as a residual. Cash is what remains of wealth after bills and bonds.

\begin{align} H_h(t) = V(t) - B_h(t) - p_{bl}(t) \cdot BL_h(t) \end{align}

  1. Interest On Bills Household

Calculate the interest earned on bills by the household.

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

  1. National Income

Calculate the national income.

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

  1. Target Proportion

Calculate the target proportion of bonds in household govt debt. The target proportion is the ratio of the value of long-term bonds to the total value of bonds and bills held by households.

\begin{align} 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)} \end{align}

  1. Taxes

Calculate the taxes. The tax base includes national income, interest on bills, and bond coupon income.

\begin{align} T(t) = \theta \cdot \left(Y(t) + r_b(t-1) \cdot B_h(t-1) + BL_h(t-1)\right) \end{align}

  1. Wealth

Calculate the wealth. Wealth evolves as savings plus capital gains on bonds.

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