Behavioral Equations IOPC#

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 \left(\frac{YD^e(t)}{p_c(t)} - \pi(t)\right) + \alpha_2 \frac{V(t-1)}{p_c(t)} \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. Final Demand

Calculate the final demand as the sum of household and government demands spread over the sectors

\begin{align} d_i(t) = \beta_{HH,i}C_{HH}(t) + \beta_{GOV,i}G(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. Inflation

Compute the inflation (i.e. term for absence of money illusion)

\begin{align} \pi(t) &= \left(\frac{p_c(t) - p_c(t-1)}{p_c(t-1)}\right)\left(\frac{V(t-1)}{p_c(t-1)}\right) \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

National income is the sum of nominal final demand

\begin{align} Y(t) = P^\top(t)d(t) \end{align}

  1. Price Indices

Compute the consumer and government price indices based on their consumption shares

\begin{align} p_c(t) &= \beta_{HH}^\top P(t)\\ p_g(t) &= \beta_{G}^\top P(t) \end{align}

  1. Prices

Compute the sectoral prices as the sum of unit labour cost and a markup on intermediate prices

\begin{align} P_i(t) = \frac{w}{pr_i} + (1 + \mu)\sum_j a_{ij}P_j(t) \end{align}

  1. Propensity To Consume Income

Endogenous propensity to consume out of income, dependent on the rate of interest

\begin{align} \alpha_1(t) = \alpha_{10} - \alpha_{11} r(t-1) \end{align}

  1. Real Gross Output

Compute real gross output as the solution to the linear set of equations

\begin{align} x(t) = (I - A)^{-1}d(t) \end{align}

  1. Set Interest Rate

Set the interest rate. This is given exogenously by the scenario.

\begin{align} r(t) = \bar{r} \end{align) \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}