Model concept, solver and details - WITCH

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Model Documentation - WITCH
Corresponding documentation
Model information
Institution European Institute on Economics and the Environment (RFF-CMCC EIEE), Italy,
Solution concept General equilibrium (closed economy)
Solution method Optimization

General Framework

WITCH (World Induced Technical Change Hybrid) is an optimal growth model of the world economy that integrates in a unified framework the sources and the consequences of climate change. A climate module links GHG emissions produced by economic activities to their accumulation in the atmosphere and the oceans. The effect of these GHG concentrations on the global mean temperature is derived. A damage function explicitly accounts for the effects of temperature increases on the economic system.

Regions interact with each other because of the presence of economic (technology, exhaustible natural resources) and environmental global externalities. For each region a forward-looking agent maximizes its own inter-temporal social welfare function, strategically and simultaneously to other regions. The inter-temporal equilibrium is calculated as an open-loop Nash equilibrium, or, a cooperative solution can also be solved by aggregating the welfare of each regions. More precisely, the Nash equilibrium is the outcome of a non-cooperative, simultaneous, open membership game with full information. Through the optimization process regions choose the optimal dynamic path of a set of control variables, namely investments in key economic variables.

WITCH is a hard-link hybrid model because the energy sector is fully integrated with the rest of the economy and therefore investments and the quantity of resources for energy generation are chosen optimally, together with the other macroeconomic variables. The model can be defined hybrid because the energy sector features a bottom-up characterization. A broad range of different fuels and technologies can be used in the generation of energy. The energy sector endogenously accounts for technological change, with considerations for the positive externalities stemming from Learning-By-Doing and Learning-By-Researching. Overall, the economy of each region consists of eight sectors: one final good, which can be used for consumption or investments, and seven energy sectors (or technologies): coal, oil, gas, wind & solar, nuclear, electricity, and bio-fuels.

Non-cooperative solution

The game theoretic setup makes it possible to capture the non-cooperative nature of international relationships. Free-riding behaviors and strategic inaction induced by the presence of a global externality are explicitly accounted for in the model. Climate change is the major global externality, as GHG emissions produced by each region indirectly impact on all other regions through the effect on global concentrations and thus global average temperature.

The model features other economic externalities that provide additional channels of interaction. Energy prices depend on the extraction of fossil fuels, which in turn is affected by consumption patterns of all regions in the world. International knowledge and experience spillovers are two additional sources of externalities. By investing in energy R&D, each region accumulates a stock of knowledge that augments energy efficiency and reduces the cost of specific energy technologies.

The effect of knowledge is not confined to the inventor region but it can spread to other regions. Finally, the diffusion of knowledge embodied in wind&solar experience is represented by learning curves linking investment costs with world, and not regional, cumulative capacity. Increasing capacity thus reduces investment costs for all regions. These externalities provide incentives to adopt strategic behaviours, both with respect to the environment (e.g. GHG emissions) and with respect to investments in knowledge and carbon-free but costly technologies.

Two different solutions can be produced: a cooperative one that is globally optimal and a decentralised, non-cooperative one that is strategically optimal for each given region (Nash equilibrium). In the cooperative solution all externalities are internalised and therefore it can be interpreted as a first-best solution. The Nash equilibrium instead can be seen as a second-best solution. Intermediate degree of cooperation, both in terms of externalities addressed and
participation can also be simulated.