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Behavioral Equations ECO3IOPC
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Step Equations
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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

.. math::
	:label: carbon_mass_nonrenewable_energy
	:nowrap:

	\begin{align}
	cen(t) &= \frac{emis(t)}{car}
	\end{align}


2. Central Bank Bill Holdings

Calculate the central bank bill holdings.

.. math::
	:label: central_bank_bill_holdings
	:nowrap:

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


3. Central Bank Money Stock

Calculate the central bank money stock.

.. math::
	:label: central_bank_money_stock
	:nowrap:

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


4. Central Bank Profits

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

.. math::
	:label: central_bank_profits
	:nowrap:

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


5. Co2 Intensity Change

The energy emission intensity decreases by a fixed percentage
each period

.. math::
	:label: co2_intensity_change
	:nowrap:

	\begin{align}
	\beta_e(t) = \beta_e(t-1) (1 - \Delta_\% \beta_e)
	\end{align}


6. Consumption

Calculate the consumption.

.. math::
	:label: consumption
	:nowrap:

	\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}


7. Cumulative Co2 Emissions

Cumulative CO2 emissions are simply incremented by the current
emissions

.. math::
	:label: cumulative_co2_emissions
	:nowrap:

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


8. Discarding Of Socioeconomic Stock

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

.. math::
	:label: discarding_of_socioeconomic_stock
	:nowrap:

	\begin{align}
	dis(t) &= m_{mat}^\top (\zeta \cdot dc(t-1))
	\end{align}


9. Disposable Income

Calculate the disposable income.

.. math::
	:label: disposable_income
	:nowrap:

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


10. Emissions From Nonrenewable Energy

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

.. math::
	:label: emissions_from_nonrenewable_energy
	:nowrap:

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


11. Energy Reserves

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

.. math::
	:label: energy_reserves
	:nowrap:

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


12. Energy To Resource Conversion

Energy resources are converted into reserves at a fixed rate

.. math::
	:label: energy_to_resource_conversion
	:nowrap:

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


13. Energy Used In Production

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

.. math::
	:label: energy_used_in_production
	:nowrap:

	\begin{align}
	en(t) = \epsilon_e^\top x(t)
	\end{align}


14. Expected Disposable Income

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

.. math::
	:label: expected_disposable_income
	:nowrap:

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


15. Expected Wealth

Calculate the expected wealth.

.. math::
	:label: expected_wealth
	:nowrap:

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


16. Extraction Of Matter

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

.. math::
	:label: extraction_of_matter
	:nowrap:

	\begin{align}
	mat(t) &= x_{mat} - rec(t)
	\end{align}


17. Final Demand

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

.. math::
	:label: final_demand
	:nowrap:

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


18. Government Bill Issuance

Calculate the government bill issuance.

.. math::
	:label: government_bill_issuance
	:nowrap:

	\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}


19. Household Bill Demand

Calculate the household bill demand.

.. math::
	:label: household_bill_demand
	:nowrap:

	\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}


20. Household Bill Holdings

Calculate the household bill holdings.

.. math::
	:label: household_bill_holdings
	:nowrap:

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


21. Household Money Stock

Calculate the household deposits as a residual.

.. math::
	:label: household_money_stock
	:nowrap:

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


22. Inflation

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

.. math::
	:label: inflation
	:nowrap:

	\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}


23. Interest Earned On Bills Household

Calculate the interest earned on bills by the household.

.. math::
	:label: interest_earned_on_bills_household
	:nowrap:

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


24. Material Goods Production

The material goods production in the economy

.. math::
	:label: material_goods_production
	:nowrap:

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


25. Matter Reserves

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

.. math::
	:label: matter_reserves
	:nowrap:

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


26. Matter To Resource Conversion

Matter resources is converted into reserves at a fixed rate

.. math::
	:label: matter_to_resource_conversion
	:nowrap:

	\begin{align}
	res(t) &= res(t-1) - conv_m(t)\\
	conv_m(t) &= \sigma_m res(t)
	\end{align}


27. National Income

National income is the sum of nominal final demand

.. math::
	:label: national_income
	:nowrap:

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


28. Non Renewable Energy Used In Production

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

.. math::
	:label: non_renewable_energy_used_in_production
	:nowrap:

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


29. Oxygen

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

.. math::
	:label: oxygen
	:nowrap:

	\begin{align}
	o2(t) &= emis(t) - cen(t)
	\end{align}


30. Price Indices

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

.. math::
	:label: price_indices
	:nowrap:

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


31. Prices

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

.. math::
	:label: prices
	:nowrap:

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


32. Propensity To Consume Income

Endogenous propensity to consume out of income, dependent on the
rate of interest and on the deviation of temperature from its
initial reference value. The reference temperature is captured at
``initialize()`` and carried forward through ``prior`` because
``Behavior.forward`` resets ``state`` at the start of each step. The
temperature read is the previous step's value, since
``temperature()`` runs after this method inside ``step``. The
propensity is clamped at zero so it cannot become negative under
extreme warming.

.. math::
	:label: propensity_to_consume_income
	:nowrap:

	\begin{align}
	\alpha_1(t) = \max\!\left(0,\ \alpha_{10} - \alpha_{11} r(t-1) - \alpha_{12}\big(\mathrm{temp}(t-1) - \mathrm{temp}(0)\big)\right)
	\end{align}


33. Real Gross Output

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

.. math::
	:label: real_gross_output
	:nowrap:

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


34. Recycling Of Discarded Stock

A fixed share of the discarded socioeconomic stock is recycled

.. math::
	:label: recycling_of_discarded_stock
	:nowrap:

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


35. 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

.. math::
	:label: renewable_energy_used_in_production
	:nowrap:

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


36. Set Interest Rate

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

.. math::
	:label: set_interest_rate
	:nowrap:

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


37. Socioeconomic Stock

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

.. math::
	:label: socioeconomic_stock
	:nowrap:

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


38. Stock Of Durable Goods

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

.. math::
	:label: stock_of_durable_goods
	:nowrap:

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


39. Taxes

Calculate the taxes.

.. math::
	:label: taxes
	:nowrap:

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


40. Temperature

Temperature is determined by a transformation of cumulative CO2

.. math::
	:label: temperature
	:nowrap:

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


41. Waste

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

.. math::
	:label: waste
	:nowrap:

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


42. Wealth

Calculate the wealth.

.. math::
	:label: wealth
	:nowrap:

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