Behavioral Equations ECO3IOPC#

Step Equations#

  1. Carbon Mass Nonrenewable Energy

The carbon mass of non-renewable energy is given by the conversion of emissions (due to non-renewable energy) with a fixed constant

\begin{align} cen(t) &= \frac{emis(t)}{car} \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(t-1)B_{CB}(t-1) \end{align}

  1. Co2 Intensity Change

The energy emission intensity decreases by a fixed percentage each period

\begin{align} \beta_e(t) = \beta_e(t-1) (1 - \Delta_\% \beta_e) \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. Cumulative Co2 Emissions

Cumulative CO2 emissions are simply incremented by the current emissions

\begin{align} co2_{cum}(t) = co2_{cum}(t-1) + emis(t) \end{align}

  1. Discarding Of Socioeconomic Stock

The discarding of socioeconomic stock occurs as a percentage of existing stock, converted into units of matter

\begin{align} dis(t) &= m_{mat}^\top (\zeta \cdot dc(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. Emissions From Nonrenewable Energy

Emissions are based on the use of non-renewable energy, with a fixed emission intensity

\begin{align} emis(t) = \beta_e nen(t) \end{align}

  1. Energy Reserves

Energyreserves are depleted by human use and incremented by the conversion from resources

\begin{align} k_e(t) &= k_e(t-1) + conv_e(t) - mat(t) \end{align}

  1. Energy To Resource Conversion

Energy resources are converted into reserves at a fixed rate

\begin{align} res_e(t) &= res_e(t-1) - conv_e(t)\\ conv_e(t) &= \sigma_e res_e(t) \end{align}

  1. Energy Used In Production

Energy use in production is given by a fixed energy intensity of production

\begin{align} en(t) = \epsilon_e^\top x(t) \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. Extraction Of Matter

The matter extracted is the difference in the matter consumed and the matter that was recycled

\begin{align} mat(t) &= x_{mat} - rec(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. Material Goods Production

The material goods production in the economy

\begin{align} x_{mat}(t) &= m_{mat}^\top x(t) \end{align}

  1. Matter Reserves

Matter reserves are depleted by human use and incremented by the conversion from resources

\begin{align} k_m(t) &= k_m(t-1) + conv_m(t) - mat(t) \end{align}

  1. Matter To Resource Conversion

Matter resources is converted into reserves at a fixed rate

\begin{align} res(t) &= res(t-1) - conv_m(t)\\ conv_m(t) &= \sigma_m res(t) \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. Non Renewable Energy Used In Production

Non-renewable energy use in production is given by the difference in energy used and renewable energy used.

\begin{align} nen(t) = en(t) - ren(t) \end{align}

  1. Oxygen

The oxygen level is given by the difference in emissions and the carbon mass of energy

\begin{align} o2(t) &= emis(t) - cen(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. Recycling Of Discarded Stock

A fixed share of the discarded socioeconomic stock is recycled

\begin{align} rec(t) &= \rho_{dis} dis(t) \end{align}

  1. Renewable Energy Used In Production

Renewable energy use in production is given by a fixed energy intensity of production combined with a fixed share of energy sourced from renewables

\begin{align} ren(t) = \epsilon_e^\top (\eta_{en} \cdot x(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. Socioeconomic Stock

The socioeconomic stock grows through material extraction and shrinks due to discards

\begin{align} k_h(t) &= k_h(t-1) + x_{mat}(t) - dis(t) \end{align}

  1. Stock Of Durable Goods

The stock of durable goods evolves based on inflows from consumption and outflows from discard

\begin{align} dc(t) &= dc(t-1) + B_c c(t) - \zeta dc(t-1) \end{align}

  1. Taxes

Calculate the taxes.

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

  1. Temperature

Temperature is determined by a transformation of cumulative CO2

\begin{align} temp(t) = \frac{1}{1-fnc}\cdot tcre \cdot co2_{cum}(t) \end{align}

  1. Waste

Waste is computed as the difference in matter extraction and the growth in the SocioeconomicStock

\begin{align} wa(t) &= mat(t) - (k_h(t) - k_h(t-1)) \end{align}

  1. Wealth

Calculate the wealth.

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