.. note:: This page is generated from inline documentation in ``MACRO/macro_core.gms``. .. _macro-core: MACRO core formulation ====================== MACRO is a macroeconomic model maximizing the intertemporal utility function of a single representative producer-consumer in each node (or macro-economic region). The optimization result is a sequence of optimal savings, investment, and consumption decisions. The main variables of the model are the capital stock, available labor, and commodity inputs, which together determine the total output of an economy according to a nested constant elasticity of substitution (CES) production function. End-use service demands in the (commercial) demand categories of MESSAGE is determined within the model, and is consistent with commodity supply curves, which are inputs to the model. Notation declaration ~~~~~~~~~~~~~~~~~~~~ The following short notation is used in the mathematical description relative to the GAMS code: ============== =============================== =================================================================== Math Notation GAMS set & index notation Description ============== =============================== =================================================================== :math:`n` node (or node_active in loops) spatial node corresponding to the macro-economic MESSAGE regions :math:`y` year year (2005, 2010, 2020, ..., 2100) :math:`s` sector sector corresponding to the (commercial) end-use demands of MESSAGE ============== =============================== =================================================================== A listing of all parameters used in MACRO together with a decription can be found in the table below. ================================== ================================================================================================================================ Parameter Description ================================== ================================================================================================================================ :math:`\text{period}_t` Number of years in time period :math:`t` (forward diff) :math:`\text{total_cost}_{n,t}` Total system costs in region :math:`n` and period :math:`t` from MESSAGE model run :math:`\text{enestart}_{n,s,t}` Consumption level of (commercial) end-use services :math:`s` in region :math:`n` and period :math:`t` from MESSAGE model run :math:`\text{p}_{n,s,t}` Shadow prices of (commercial) end-use services :math:`s` in region :math:`n` and period :math:`t` from MESSAGE model run :math:`\text{E}_{min,n,s,t}` Subsistence level of direct energy consumption (end-use service) in region :math:`n`, sector :math:`s` and period :math:`t` :math:`\text{h}_{n,s,t}` Share of the direct energy consumption of the total energy production in region :math:`n`, sector :math:`s` and period :math:`t` :math:`\epsilon_n` Elasticity of substitution between capital-labor and total energy in region :math:`n` :math:`\rho_n` :math:`\epsilon - 1 / \epsilon` where :math:`\epsilon` is the elasticity of substitution in region :math:`n` :math:`\beta_n` Consumption value share parameter in region :math:`n` :math:`\sigma_{n,s}` Direct energy consumption value share parameter in region :math:`n` and of sector :math:`s` :math:`\delta_n` Annual depreciation rate in region :math:`n` :math:`\alpha_n` Capital value share parameter in region :math:`n` :math:`a_n` Production function coefficient of capital and labor in region :math:`n` :math:`b_{n,s}` Production function coefficients of the different end-use sectors in region :math:`n`, sector :math:`s` and period :math:`t` :math:`\text{udf}_{n,t}` Utility discount factor in period year in region :math:`n` and period :math:`t` :math:`\text{L}_{n,t}` Labor force in region :math:`n` and period :math:`t` :math:`\text{w}_{n,t}` Wage rate in region :math:`n` and period :math:`t` :math:`\text{grow}_{n,t}` Annual growth rates of potential GDP in region :math:`n` and period :math:`t` :math:`\text{aeei}_{n,s,t}` Autonomous energy efficiency improvement (AEEI) in region :math:`n`, sector :math:`s` and period :math:`t` :math:`\text{fin_time}_{n,t}` Finite time horizon correction factor in utility function in region :math:`n` and period :math:`t` ================================== ================================================================================================================================ Decision variables ~~~~~~~~~~~~~~~~~~~~ =============================== =========================================================== ============================================================================================================== Variable Definition Description =============================== =========================================================== ============================================================================================================== :math:`\text{K}_{n,y}` :math:`\text{K}_{n, y}\geq 0 ~ \forall n, y` Capital stock in region :math:`n` and period :math:`y` :math:`\text{Y}_{n,y}` :math:`\text{Y}_{n, y}\geq 0 ~ \forall n, y` Total production in region :math:`n` and period :math:`y` :math:`\text{C}_{n,y}` :math:`\text{C}_{n, y}\geq 0 ~ \forall n, y` Consumption in region :math:`n` and period :math:`y` :math:`\text{PHYSENE}_{n,s,y}` :math:`\text{PHYSENE}_{n, s, y}\geq 0 ~ \forall n, s, y` Physical end-use service use in region :math:`n`, sector :math:`s` and period :math:`y` :math:`\text{TE}_{n,s,y}` :math:`\text{TE}_{n, s, y}\geq 0 ~ \forall n, s, y` Value of total end-use service in the production function and utility function in region :math:`n`, sector :math:`s` and period :math:`y` :math:`\text{E}_{n,s,y}` :math:`\text{E}_{n, s, y}\geq 0 ~ \forall n, s, y` Value of direct energy consumption of end-use service of households in the utility function in region :math:`n`, sector :math:`s` and period :math:`y` :math:`\text{YE}_{n,s,y}` :math:`\text{YE}_{n, s, y}\geq 0 ~ \forall n, s, y` Value of end-use service energy consumption in the production function in region :math:`n`, sector :math:`s` and period :math:`y` :math:`\text{UTILITY}` :math:`\text{UTILITY} \in \left[-\infty..\infty\right]` Utility function (discounted log of consumption) =============================== =========================================================== ============================================================================================================== Equation UTILITY_FUNCTION --------------------------------- The utility function, which is maximized, sums the discounted logarithm of consumption :math:`\text{C}_{n,y}` and direct energy consumption of end-use services :math:`\text{E}_{n,s,y}` of a single representative household over the entire time horizon of the model. The utility function and the capital formulation of the optimization problem are derived in previously (See equations 10 and 13 of the model documentation). .. math:: \text{UTILITY} = \sum_{n} \bigg( & \sum_{y | ( ( {ord}( y ) > 1 ) \wedge ( {ord}( y ) < | y | ) )} \text{udf}_{n, y} \cdot \bigg( (\beta_n + \sum_{s=1}^{3} \sigma_{s,n}) \log(\text{C}_{n, y}) - \sum_{s=1}^{3} \sigma_{s,n} \log(\text{p}_{n,s,y}) + \sum_{s=1}^{3} \sigma_{s,n} \log\left(\frac{\sigma_{s,n}}{\beta_n}\right) \bigg) \cdot \text{duration_period}_{y} \\ + &\sum_{y | ( {ord}( y ) = | y | ) } \text{udf}_{n, y} \cdot \bigg( (\beta_n + \sum_{s=1}^{3} \sigma_{s,n}) \log(\text{C}_{n, y}) - \sum_{s=1}^{3} \sigma_{s,n} \log(\text{p}_{n,s,y}) + \sum_{s=1}^{3} \sigma_{s,n} \log\left(\frac{\sigma_{s,n}}{\beta_n}\right) \bigg) \cdot \big( \text{duration_period}_{y-1} + \frac{1}{\text{fin_time}_{n, y}} \big) \bigg) Equation CAPITAL --------------------------------- The household maximizes its utility subject to the constraint on wealth accumulation of capital in the sectors not represented in the energy model MESSAGE. The net capital (or wealth) formation :math:`\text{K}_{n,t}` is derived from the existing capital stock, returns on capital, labor income, minus the expenses for direct energy consumption and all other consumption goods, as well as depreciation of the previous capital stock. .. math:: \text{K}_{t+1,n} = (1 - \delta_{n})^{\text{period}_{t}} \text{K}_{t,n} + ((1 + r_{t,n})^{\text{period}_{t}} - 1) \text{K}_{t,n} + \text{period}_{t} \left( w_{t,n} L_{t,n} - \sum_{s=1}^{3} p_{t,s,n} L_{n,t} E_{min,t,s,n} - \frac{\beta + \sum_{s=1}^{3} \sigma_{s,n}}{\beta} C_{t,n} \right) \qquad (15) Equation PRODUCTION --------------------------------- We implement a nested constant elasticity of substitution (CES) production function with capital, labor, and the (commercial) end-use services represented in MESSAGE as inputs. :math:`\text{Y}_{n,t}` should correspond to gross domestic product (GDP). .. math:: \text{Y}_{n,t} = \left( a_{n} \cdot \text{K}_{n, t}^{ ( \rho_{n} \cdot \alpha_{n} ) } \cdot \text{L}_{n, t}^{ ( \rho_{n} \cdot ( 1 - \alpha_{n} ) ) } + \sum_{s} ( b_{n, s} \cdot \text{YE}_{n, s, t}^{\rho_{n}} ) \right)^{ \frac{1}{\rho_{n}} } \qquad \forall n, t > 1 \qquad (16) Equations NEW ENERGY ACCOUNTING --------------------------------- ew energy accounting equations. Need to be discussed and checked. See model documentation.