# Introduction

This Appendix contains the documentation on the Damage Cost Function extensions for the TIMES model. The chapter contains 6 sections: section {numref}`%s <b-mathematical-formulations>` contains the mathematical formulation, section {numref}`%s <b-switches-and-parameters>` describes the parameters for the Damage Cost Functions, and section {numref}`%s <b-examples>` gives two examples. Finally, section {numref}`%s <b-variables>` describes the variables and section {numref}`%s <b-equations>` describes the equations.

The Damage Cost Function option of TIMES is intended for modelers who wish to evaluate the environmental externalities caused by an energy system. For instance, emissions of toxic or environmentally harmful pollutants from the energy system create social costs linked to impacts of the pollution on human health and the environment. In another example, in global studies of GHG emissions, it may be of interest to evaluate the impact of GHG emissions on concentrations and ultimately on damages created by climate change induced by increased concentration of GHGs.

Until recently, in most studies involving bottom-up models, emission externalities have been modeled in one of two ways: either by introducing an emission tax, or by imposing emission caps. In the first case, the tax is (ideally) supposed to represent the external cost created by one unit of emission. However, using a tax assumes that the cost is a linear function of emissions. In the second approach, it is assumed that such a cost is unknown but that exogenous studies (or regulations, treaties, etc.) have defined a level of acceptable emissions that should not be exceeded. However, using this approach is akin to making the implicit assumption that emissions in excess of the cap have an infinite external cost. Both of these approaches have merit and have been successfully applied to many energy system model studies.

It is however possible to extend these two approaches by introducing an option to better model the cost of damages created by emissions. The damage function option discussed in this document extends the concept of an emission tax by modeling more accurately the assumed cost of damages due to emissions of a pollutant.