## 3.1 Welfare function

In WITCH, a social planner maximises the sum of regional discounted utility \(W\) of each coalition, \(clt\). The regional utility function at any point in time and each region is based on a power or constant relative risk aversion (CRRA) utility function derived from consumption per capita. For different coalitions, the non-cooperative solution, and the globally optimal mode, the objective function can take different forms based on the implied set of coalitions \(clt\).

### 3.1.1 Utility function

The WITCH model can be run in a non-cooperative mode, where each region \(n\) acts as one player maximising its welfare. In this case, the set of players or coalitions \(clt\) consists each of a single region. For each region \(n\) and time period \(t\), intertemporal utility is computed as discounted sum of utility (taking into account the region’s population \(l(t,n)\)) based on a utility function with a degree of relative risk aversion of \(\eta\):

\[ W(n) = \sum_{t} l(t,n) \frac{\left(\frac{C(t,n)}{l(t,n)}\right)^{1-\eta}-1}{1-\eta} \beta^{t}. \]

\(C\) is total consumption, \(l\) is population, and the pure time preference discount factor \(\beta\) is given by the standard geometric discounting rule:

\[ \beta = ({1+\rho})^{-\Delta_{t}} \]

where \(\Delta_{t}=5\) is the duration of one time step in years and \(rho\) the discount rate.

### 3.1.2 Welfare of coalitions

In the cooperative mode, different coalitions can be formed including the Grand Coalition containing all regions. With cooperation, there are hence one or several coalitions \(clt\), who maximise total welfare in their member region(s). The coalition \(clt\) maximises a sum of the welfare of the member regions. There are several welfare concepts admissible to aggregate welfare across coalition members.

The default option is a (disentangled) Utilitarian solution, taking into account inequality across regions through a degree of inequality aversion based on equity equivalents, which is described also in (Berger and Emmerling 2017). This welfare concept is related to Epstein-Zin preferences (Epstein and Zin 1989) and welfare of a coalition \(clt\) is given by the equation:

\[ W(clt)= \sum_{t=1}^{T} \left(\sum_{n \in clt}l(t,n)\right) \left[\frac{1}{1-\eta}\left( \frac{1}{\sum_{n \in clt}l(t,n)} \sum_{n \in clt}l(t,n) \left(\frac{C(t,n)}{l(t,n)} \right )^{1-\gamma}\right)^{\frac{1-\eta}{1-\gamma}}-1\right] \beta^t \]

Note that by default, we consider no inequality aversion(\(gamma=0\)), since this equalises marginal welfare across regions, leads to a unique social cost of carbon, and is quantitatively similar to the common use of Negishi weights (Negishi 1960). As an alternative, time varying Negishi weights (Nordhaus and Yang 1996) can be used in which case the welfare function has the form of

\[ W^{Negishi}(clt) = \sum_{t}\sum_{n\in clt} w_{t,n} l(t,n) \frac{\left(\frac{C(t,n)}{l(t,n)}\right)^{1-\eta}-1}{1-\eta} \beta^t\]

where \(w_{t,n}\) are the time- and region-specific Negishi weights which are computed as

\[ w_{t,n}=\frac{\frac{1}{c(t,n)^{\eta}}}{\sum_{n' \in clt}\frac{1}{c(t,n')^{-\eta}}}. \]

For the current version of WITCH, the parameters are based on an overview of recent contributions in the literature on discounting, with an intermediate parametrizations of a one per-cent utility discount rate and a degree of relative risk aversion of 1.5. The full parametrization is reproduced in Table 1.1.

Symbol | Definition | GAMS | Default value |
---|---|---|---|

\(\eta\) | Inverse of IES | eta | 1.50 |

\(\rho\) | Pure rate of time preference | srtp(t) | 0.01 |

\(\gamma\) | Degree or inequality aversion | gamma | 0.00 |

### References

Berger, Loic, and Johannes Emmerling. 2017. “Welfare as Simple(x) Equity Equivalents.” 2017.014. Milan: Fondazione Eni Enrico Mattei. http://www.feem.it/getpage.aspx?id=9013.

Epstein, Larry G., and Stanley E. Zin. 1989. “Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework.” *Econometrica* 57 (4): 937–69.

Negishi, Takashi. 1960. “Welfare Economics and Existence of an Equilibrium for a Competitive Economy.” *Metroeconomica* 12 (2-3). Wiley-Blackwell: 92–97.

Nordhaus, William D., and Zili Yang. 1996. “A Regional Dynamic General-Equilibrium Model of Alternative Climate-Change Strategies.” *The American Economic Review* 86 (4): 741–65.