## 3.2 The general economy

### 3.2.1 Consumption

The single argument of the utility function is per-capita consumption of the representative agent in each region. Total consumption \(C\) is thereby defined by the budget constraint, whereby from net output \(Y\), see next section for a definition, the following investments, and operation and maintenance costs (computed as coefficients of the installed capacity) are subtracted:

final good investments, \(I_{FG}\),

investments in energy technology \(j\), \(I_j\) and

research and development investments in energy technology \(j\), (R&D), \(I_{RD,j}\),

extraction sector investments, \(I_{OUT,f}\)

infrastructure for the electric grid investments, \(I_{GRID}\)

operation and maintenance costs in energy technology \(j\), \(oem(j)\) and

extraction costs of fuel \(f\), \(oem\_ex(f)\).

Investments in adaptation (\(I(PRADA,t,n)\), \(I(SCAP,t,n)\), \(I(RADA,t,n)\))

\[\begin{align*} C(t,n) = & Y(t,n) \\ & - I_{FG}(t,n) \\ & - \sum_j\left( I_{RD,j}(t,n) + I_j(t,n) + (oem_j(t,n)*K_j(t,n)) \right) \\ & - \sum_f \left( I_{OUT,f}(t,n) + (oem\_ex_f*Q_{OUT,f}(t,n)) \right) \\ & - I_{GRID}(t,n) \\ & - I(PRADA,t,n) - I(SCAP,t,n) - I(RADA,t,n) \\ \end{align*}\]

where

Energy technologies are denoted \(j\)

Fossil fuels are denoted \(f\)

### 3.2.2 Output

The production side of the economy is very aggregated. Each region produces one single commodity that can be used for consumption or investments. The final good \(Y\) is produced via a nested CES function that combines capital(\(K\)), labour (\(L\)) and energy services (\(ES\)). Capital and lobar are aggregated using a Cobb-Douglas production function. This output aggregate is then combined with energy services with a CES production function. The climate impacts \(\Omega(t,n)\) affect gross output, so that a certain share of output is lost due to climate change impacts. Moreover, the costs of fossil fuels, \(C_f\), are subtracted from gross output. Also, costs related to mitigation of GHG emissions \(C_{ghg}(t,n)\) are subtracted here: they include costs of direct emission mitigation including Carbon Capture and Storage (CCS), the costs of avoided deforestation and degradation (REDD+), the carbon tax revenues, and, in the case of a permit trading scheme, the net imports of permits. Net output is thus obtained as

\[\begin{align*} Y(t,n) = & \frac{ tfp0(n) \left( \alpha(n) \left( tfpy(t,n) K_{FG}(t,n)^{\beta(n)}l(t,n)^{(1-\beta(n))} \right)^{\rho} + (1-\alpha(n)) ES^{\rho}(t,n) \right)^{\frac{1}{\rho}} }{\Omega(t,n)} \\ &- \sum_f C_f(t,n)\\ &- \sum_{ghg} C_{ghg}(t,n), \end{align*}\]

where the CES parameters are \(\alpha(n)\) and \(\rho\). The parameter \(\rho\) is computed such that \(\rho=\frac{\mathcal{s}-1}{\mathcal{s}}\), where \(\mathcal{s}\) is the elasticity of substitution. The parameter \(\beta(n)\) represents the Cobb-Douglas coefficient of the capital-labour aggregate.

Total factor productivity \(tfpy(t,n)\) is dynamically calibrated and evolves exogenously with time. Labour \(l\) is assumed to be equal to to population, thus assuming no unemployment. Finally, the parameter \(tfp0(n)\) is calibrated to match GDP in the base year.

### 3.2.3 Capital

The capital stock in the final good sector accumulates following the standard capital accumulation rule with exponential depreciation and an annual depreciation rate of \(\delta_{FG}=10%\):

\[ K_{FG}(t+1,n) = K_{FG}(t,n)\times(1-\delta_{FG})^{\Delta_{\text{t}}} + \Delta_{\text{t}}\times I_{FG}(t,n) \]

### 3.2.4 Energy Services

Energy services \(ES\) are provided by a combination of physical energy input and a stock of energy efficiency knowledge. This allows for endogenous improvements in energy efficiency. Energy efficiency can be increased through investments in energy R&D, which build up the stock of knowledge. The stock of knowledge can then substitute physical energy in the production of energy services. More details on the stock of knowledge are available in Research and Development.

Energy services \(ES\) are an aggregate of the amount of energy consumed, \(EN\), and a stock of knowledge, \(RDEN\), combined within a CES function:

\[ ES(t,n) = \phi_{ES}(n)\left( \alpha_{ES}(n) RDEN(t,n)^{\rho_{ES}} + (1-\alpha_{ES}(n)) tfpn(t,n) EN(t,n)^{\rho_{ES}} \right)^{\frac{1}{\rho_{ES}}} \]

The CES parameters \(\phi_{ES}(n)\), \(\alpha_{ES}(n)\), and \(\rho_{ES}\) are statically calibrated for each region based on prices and quantities in the base year 2005. The factor productivity of energy \(tfpn(t,n)\) calibrated is presented in the Dynamic calibration section. The elasticities including \(\rho_{ES}\) are chosen to fit empirical substitution between different energy uses and fuels, notably referring to (Koetse, Groot, and Florax 2008) and (Stern 2012a). The elasticity between electric and non-electric energy of \(0.66\) has been chosen to match the estimates of (Koetse, Groot, and Florax 2008).

### 3.2.5 Energy

Energy used in the economy is a combination of electricity and non-electric energy. Electric energy can be generated using a set of different technology options. Non-electric energy comprises the use of different fuels, namely coal, gas, oil, biomass, and a backstop technology for industry, residential households, and transportation. The aggregation uses a CES function with parameters \(\alpha_{EN}(n)\) and \(\rho_{EN}\).

\[ EN(t,n) = \left( \alpha_{EN}(n) EL(t,n)^{\rho_{EN}} + (1-\alpha_{EN}(n)) NEL(t,n)^{\rho_{EN}} \right)^{\frac{1}{\rho_{EN}}} \]

Each factor is further decomposed into several sub-components. The components are aggregated using CES, linear and Leontief production functions, which is described in detail in the Energy module.

### References

Koetse, Mark J., Henri L. F. de Groot, and Raymond J. G. M. Florax. 2008. “Capital-Energy Substitution and Shifts in Factor Demand: A Meta-Analysis.” *Energy Economics* 30 (5): 2236–51. https://doi.org/10.1016/j.eneco.2007.06.006.

Stern, David I. 2012a. “Interfuel Substitution: A Meta-Analysis.” *Journal of Economic Surveys* 26 (2): 307–31. https://doi.org/10.1111/j.1467-6419.2010.00646.x.