# Energy and CO2 Emissions (1.1)

The value of energy aggregate in sector $j$ in country $r$ at time $t$ is denoted by $ETOT_{j,r,t}$. It gathers electricity and the energy other than electricity made from fossil fuels.

This CES energy aggregate is subject to productivity improvements, $EE_{j,r,t}$. The dynamic of the energy productivity is projected by the growth model on which the dynamic baseline is based. The productivity parameter is introduced in the FOC conditions that define the demand for total energy, based on the CES bundle made of capital and energy , $KE_{j,r,t}$. The FOC is then given by :

$$ETOT_{j,r,t} = a_E EE_{j,r,t} KE_{j,r,t} \left( \frac{PKE_{j,r,t}}{PE_{j,r,t}}\right)^{\sigma_{KE}}$$

Using CES functional forms with variables in monetary units leads to inconsistencies when trying to retrieve physical quantities. In our case, this matters for energy consumption, production, and trade, and their consequences for CO2 emissions1). Therefore, in addition to accounting relations in constant dollars, MIRAGE-e integrates a parallel accounting in energy physical quantities (in million tons of oil equivalent, Mtoe) allowing CO2 emissions to be computed (in million tons of carbon dioxide, MtCO2). Since the CES architecture does not maintain coherence in physical quantities, MIRAGE-e introduces energy- and country-specific adjustment coefficients. These two aggregation coefficients allow our basic energy accounting relationships to remain valid. This means that the quantity produced by one country $EY_{e,r,t}$ must equal the demand in this country both local, $ED_{e,r,t}$ and from abroad, $EDEM_{e,r,s,t}$ ; and energy consumption (by households, $EC_{e,s,t}$ and firms, $EEIC_{e,j,s,t}$) in one country must equal its local and foreign demand.

$$EY_{e,r,t} = ED_{e,r,t} + \sum_{s} EDEM_{e,r,s,t}$$ $$EC_{e,s,t} + \sum_j EEIC_{e,j,s,t} = ED_{e,s,t} + \sum_r EDEM_{e,r,s,t}$$

The corresponding adjustment coefficient, $AgDem_{e,r,t}$ (resp. $AgCons_{e,r,t}$) rescales the country's demand (resp. consumption) such that it matches the physical quantities produced (resp. demanded). In turn, only energy quantity produced is proportional to the volume production $Y$ due to its being above rather than inside the CES. The epsilons below are constant conversion coefficients calibrated from the energy quantity data; they allow us to link energy quantities with corresponding volumes of demand for local good, $D_{e,r,t}$, bilateral demand, $DEM_{e,r,t}$, local final consumption, $C_{e,s,t}$ and local intermediate consumption, $EIC_{e,j,s,t}$.

$$EY_{e,r,t} = \epsilon_{e,r}^{Y} Y_{e,r,t}$$ $$ED_{e,r,t} = \epsilon_{e,r}^{D} AgDem_{e,r,t} D_{e,r,t}$$ $$EDEM_{e,r,s,t} = \epsilon_{e,r,s}^{DEM} AgDem_{e,r,t} DEM_{e,r,s,t}$$ $$EC_{e,s,t} = \epsilon_{e,s}^{C} AgCons_{e,s,t} C_{e,s,t}$$ $$EEIC_{e,j,s,t} = \epsilon_{e,j,s}^{EIC} AgCons_{e,s,t} EIC_{e,j,s,t}$$

Finally, CO2 emissions are recovered as proportional to the energy quantities consumed, using energy-, sector- and country-specific factors determined by the data.

1) Preliminary simulations of MIRAGE-e showed that there could be a gap of more than 20% between a country’s energy consumption and energy demanded if proportionality was assumed between monetary and physical values.