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### Climate Treaty

You have the option of simulating a climate treaty by setting carbon caps in three years, 2050, 2100 and 2150. The caps are a specified percentage of emissions in 2005 and apply in each time period until the next cap. For example, if the cap for 2050 is 100%, emissions in each year between 2050 and 2100 are capped at the 2005 amount.

You can also choose what portion of the world has agreed to the treaty by setting the participation fraction. As participation goes down, the countries subject to the treaty have to reduce more aggressively to meet the cap, increasing costs. more

The model implements the caps by solving for $\mu(t)$ to produce the specified level of emissions. Specifically, the model induces the controlled emissions rate $\mu(t)>0$ from the carbon caps set by users whenever emissions of a year exceeds the cap in the threshold years of 2050–2100, and 2150–$Tmax$. That is, given the equation for industrial emissions $E_{ind}(t)=\sigma(t)[1-\mu(t)]\cdot A(t)K(t)^{\gamma}L(t)^{1-\gamma},$ we set $\mu(t)$ $\mu(t)=\left\{ \begin{array}{ccc} 0 & if & E_{ind}(t) < E_{ind}(2005)\times ecap_{(date)}\\ 1-\frac{E_{ind}(2005)\times ecap_{(date)}}{\sigma(t)A(t)K(t)^{\gamma}L(t)^{1-\gamma}} & if & E_{ind}(t)>E_{ind}(2005)\times ecap_{(date)} \end{array}\right.$ where the caps are denoted by $ecap_{2050}$, $ecap_{(2100)}$, $ecap_{(2150)}$, respectively.

For instance, consider the year 2070. This year should obey the emissions cap of the year 2050. Thus $\mu(2070)=1-\frac{E_{ind}(2005)\times ecap_{(2050)}}{\sigma(2070)A(2070)K(2070)^{\gamma}L(2070)^{1-\gamma}}$ if $E_{ind)}(2070)>E_{ind}(2005)\times e2050cap$. If we have more emissions than the chosen amount for a given year, we would like to decrease it by controlling the reduction rate of $\mu(t)$.

One drawback is that the model uses step function setting caps in each threshold year and assuming that those caps apply for future years, until another threshold year comes. Abrupt reduction of emissions might be more costly to implement than smooth reduction. The step function, however, reflects common practice in treaties which specify target years.