### Abatement costs

Reducing emissions will cost money. These costs are called abatement costs. If you run the model under the default, business as usual mode, these are zero as emissions are allowed to go uncontrolled. If instead you impose a climate policy, either by setting a tax or a treaty or by allowing the computer to optimize, these costs will be positive.

The model specifies abatement costs as a fraction of the total output: this is the portion of GDP that we spend replacing fossil fuel energy with clean energy. The amount is determined by the cost of clean energy (called the backstop technology link), the extent of emissions reduction (link to emissions control rate) and how many countries or industries participate (link to the participation rate). In general, abatement costs increase with the extent of the reductions, go down the more countries and industries that participate in reductions, and go down as the costs of clean energy go down. You cannot change abatement costs directly. Instead, you can change each of these three components that make up abatement costs. more

### Backstop Technology

In the business as usual scenario, there are no controls on emissions. The model runs based on the default parameters, or, if you change the defaults, your chosen parameters. Business as usual scenarios can be viewed as telling us what might happen if we do not do anything to control emissions.

### BEAM simplified model

The default carbon model in webDICE runs quickly and is reasonable accurate over short periods of time. Over longer periods, however, it is less accurate. The reason is that as carbon concentrations in the ocean go up, its ability to absorb additional carbon goes down. The default model does not account for this eect and, therefore, over long periods of time, treats the ocean as absorbing more CO2 than it really will. The BEAM model includes a simpli ed version of ocean chemistry to account for this eect and is more accurate over longer time periods. The cost of choosing BEAM is that webDICE will run more slowly, particularly in optimization mode, where the model may time out. more

### Capital

Capital — machines, land, patents, and so forth — is used to produce economic output. The model starts with a specified level of capital based on the current economy. This capital depreciates due to wear and tear and obsolescence but in each period, the people in the model save a portion of their output, adding to existing capital. Therefore, the capital available in a given period is the capital from previous time-periods, reduced by depreciation, plus savings. more

### Carbon Mass in the Atmosphere

The carbon mass in the atmosphere is the total amount, by mass, of carbon in the atmosphere in a given period. It includes both naturally occurring carbon and carbon emitted through the use of fossil fuels. Its value is determined by the carbon model. Carbon mass in the atmosphere can be converted to the parts per million of carbon using the ratio 1 ppm = 2.13 Gt of carbon.

### Carbon mass in the lower oceans

Carbon mass in the lower oceans is the total carbon stored in the lower layers of the ocean. Its value is calculated in webDICE through the carbon model.

### Carbon mass in the upper oceans

Carbon mass in the upper oceans is the total carbon stored in the upper layer of the ocean. Its value is calculated in webDICE through the carbon model.

### Carbon tax

A carbon tax is a tax on emissions. Without a tax or some other price on emission, people do not consider the full consequences of their actions. They can emit CO2 without considering that this harms others. Economists call this a market failure: the price of fossil fuel energy does not represent the true price, unlike with most other products in the economy. A a result, people use too much fossil fuel energy.

A carbon tax fixes this problem by enforcing people to consider the full costs of fossil fuel energy, including the harms that using fuel energy will cause due to climate change. A carbon tax is thought by many to be the most efficient method of controlling emissions.

There are two ways in the model to impose a carbon tax. The first is to choose a carbon tax explicitly through the simulated carbon tax option. The second is to choose the optimization option. If you choose optimization, the computer calculates an emissions path that maximizes the objective function and then determines the carbon tax that is necessary to achieve that path. more

### Clean energy costs

To reduce emissions, we have to replace fossil fuels with other sources of energy that do not produce emissions, such as solar or wind power. Replacing fossil fuels with clean energy will cost money, and tis price determines how much it will cost to stop climate change. The model estimates the cost of replacing a portion (or all_ of the fossil fuel system starting with an estimate of the cost of replacing the entire fossil fuel system with clean energy. This price is set a relatively high level for 2005 of USD1,260/ton C or equivalently, USD344/ton CO2. The price is assumed to go down over time as our technology improves.

You can choose how fast the price of clean energy goes down. The initial rate is 5% per year so that if clean energy costs USD1,000/ton C this year, it will cost USD950/ton C next year. If you are optimistic about future technology, you can increase the rate at which the price goes down by moving the slider to _ _ _ and if you are pessimistic, you can reduce the rate, all the way down to _ _ _.

### Climate Change

The term climate change or global warming refers to the effects of emissions of greenhouse gases into the atmosphere. Greenhouse gases act as a blanket and warm the surface of the earth. As we put more greenhouse gas in the atmosphere, we in effect put on a thicker blanket, warming the planet more. The primary greenhouse gas is carbon dioxide, which we emit when we use of fossil fuels.

The basic science behind climate change has been understood for more than 100 years. It is the same science that is used to explain why the mMoon is cold, the Earth is the right temperature to support life, and Venus is too hot. The Earth’s (and the moon’s and Venus’s) temperature is determined by the relationship between the energy the Earth absorbs from the Sun and the energy it emits back. These two have to be in balance for the temperature to remain stable. Because the Sun is hot, most of its energy is in the form of visible and near-infrared light. The Earth is much cooler, so the energy it emits back into space is has longer wavelengths; it is mostly in the infrared region of the spectrum. Without an atmosphere, the resulting balance of incoming and outgoing energy would mean that the average temperature of the Earth's surface would be about -20°C (which is -4°F) — too cold to support life. The reason the Earth is not actually this cold is that it is blanketed by the atmosphere. The atmosphere is nearly transparent to incoming solar radiation but it absorbs the infrared radiation coming from the Earth. In a sense, it acts like planetary insulation. The effect is to warm the Earth by nearly 35 °C, to an average temperature of around 15°C.

The absorption of infrared radiation is due to only minor elements in the atmosphere — the greenhouse gases. The major components of the atmosphere, nitrogen and oxygen, are as transparent to infrared radiation as they are to visible light. The most abundant/significant human-caused greenhouse gas is carbon dioxide. It is only a trace element in the atmosphere, comprising now only about 395 parts per million (ppm), but it has strong effects.

Carbon dioxide occurs naturally in the atmosphere. Pre-industrial concentrations were about 280 ppm. Over the last century or so, however, humans significantly increased the concentration of carbon dioxide, largely through burning fossil fuels, land use change, and agriculture. When we burn fossil fuels, we inevitably emit the carbon in these fuels in the form of carbon dioxide. Roughly 60 percent of the annual emissions of greenhouse gases on a global basis come from fossil fuels (and 80 percent of U.S. emissions). Land use change alters the concentration of carbon dioxide in the atmosphere for many reasons. The most important reason is that trees absorb carbon dioxide through photosynthesis. When we cut down trees, we eliminate this carbon sink. In addition, if we burn the timber or it decomposes naturally, we release the carbon that they stored. A little more than 18 percent of global emissions are from forestry practices. Most of the remainder is from agriculture (13.5 percent globally), which produces emissions from the use of fertilizer and by releasing carbon stored in the soil.

The Intergovernmental Panel on Climate Change has attempted to summarize the core scientific findings, and they have explanations of the basic ideas as well as detailed discussions of the most recent scientific results. For those who want to understand the core issues, we highly recommend the IPCC’s FAQ on climate change.

For those interested in an accessible discussion of the most recent scientific papers, we recommend realclimate.org.

### Climate model

webDICE includes a model of how emissions of CO2 affect temperatures. The model has two components. The first is how carbon moves around the earth, between the oceans and the atmosphere. This component determines long emissions of CO2 stay in the atmosphere. The second is how the emissions that stay in the atmosphere increase temperatures. This latter component is called radiative forcing.

The model of how emission move around the earth assumes that emissions from economic activity are released into the atmosphere. Some of these emissions are absorbed by the upper layers of the ocean and from there into the lower ocean. The rate of absorption into the upper and lower ocean determines how long emissions stay in the atmosphere. There are no parameters choices for the climate model, but in Advanced Move, you can choose the default model, which is used in Nordhaus’s DICE models, or the ‘BEAM, simplified‘ model, which is a more accurate but slightly slower running model. (Optimization using BEAM may be particularly slow and possibly time out.) more

### Climate sensitivity

Climate sensitivity determines how much temperatures go up for a given amount of carbon dioxide in the atmosphere. It is set based on the temperature increase when the concentration of CO2 doubles from its pre-industrial level. The default value is 3.2 which means that if the concentration of CO2 doubles from its pre-industrial level of 280 parts per million (ppm) to 560 ppm, temperature will eventually increase by 3.2°. The climate sensitivity is highly uncertain. You can set it to anywhere from an optimistic value such as 1 (the temperatures will not increase very much) to a pessimistic value such as 5 (temperatures will increase drastically).

### 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

### Consumption per capita

Consumption per capita is total consumption in a period divided by the number of people in that period.

### Damages from climate change

The results in webDICE, with the exception of the social costs of carbon do not depend on the discount rate. Instead, webDICE calculates the change in consumption in each period, translates that to a change in utility via the utility function and sums up the changes in utility.

The ‘pure rate of time preference’ is subtly different than the discount rate. The discount rate applies to money or, in webDICE, things money can buy like consumption. It is like the interest rate you get when you put money in the bank or that you pay when you borrow money. The pure rate of time preference is part of the process of adding up the utility of people to determine how well society is doing. It applies directly to utility, not to money. It is a way of computing the present value of the utility of people who live in the future. more

### Depreciation

Assets, such as machines and buildings are subject to wear and tear and their value goes down over time. Depreciation represents the annual decline in value of assets in the economy. The default value is 10%. You can adjust the depreciation rate between 8% and 20%.

### Discount rate

The discount rate is used to determine how much we have to save today to produce an amount in the future. For example, if the discount rate is 10% and we want to have $110 next year, we need to save$100 today because $100 will grow into$110 if invested at 10% for one year. The structure of webDICE has an ‘implied’ discount rate: the discount rate is not specified explicitly in the model but instead can be computed from the growth rate in particular model run, the elasticity of the marginal utility of consumption, and the pure rate of time preference.

### Economic model

webDICE uses an economic model in which labor and capital combine to produce output. The productivity of the economy determines how much output can be produced with a given amount of labor and capital. It is a measure of how skilled our workers are and how good the machines are that they use. The model treats the entire world as a single economic region and therefore does not include differences among countries or trade.

These three factors - labor, capital, and productivity - change over time in the model to produce a dynamic economy. The population is assumed to grow over time, increasing the supply of labor. Productivity grows over time but the growth rate slows down. Capital in a given time period is equal to the capital that was available in the last period reduced by wear and tear, plus savings in the last period. (see capital.)

Economic output is reduced by the harms from climate change (see Damages) and by an expenditures spend on reducing emissions (see abatement). What remains can be used by people to consume or to save for the future. more

See utility.

### Emissions control rate

The emissions control rate is the fraction of emissions that are reduced or controlled by a climate change policy. For example, if a climate change policy limits emissions to 60% of what they otherwise would have been, the control rate is 40%. Under the business as usual scenario there are no controls on emissions of CO2, or the emissions control rate is set to zero. If you choose to impose a carbon tax or a treaty, or if you allow the computer to determine the optimal level of emissions reductions, the emissions control rate will be positive as some portion of emissions will no longer occur either because of the tax or because it is prohibited by the treaty. When the emissions control rate is equal to 1, emissions have ceased altogether and there is no further climate change. As we reduce emissions more, costs are assumed to go up: we choose the lowest cost reductions first and then move to higher and higher cost reductions. You can choose the rate at which these costs go up by choosing a value for abatement costs, with higher values meaning that costs go up faster as we reduce more. more

### Emissions from land-use change

Forests absorb CO2 as trees and other plants grow. If we cut down or burn down forests, we reduce the amount of CO2 that they absorb, leaving more CO2 in the atmosphere. We can treat the additional CO2 as the equivalent of emissions. webDICE includes these emissions but they contribute only a limited amount overall. more

### Energy intensity/carbon intensity

Economic activity, such as heating, transportation and manufacturing, requires energy. The major source of energy in modern economics is from fossil fuels, coal, petroleum and natural gas. This means that economic activity produces emissions, called ‘industrial emissions’ in webDICE. The richer we are, the greater the emissions, and conversely, the poorer we are, the lower the emissions.

Over time, the economy has gotten more efficient: we are able to produce the same amount using less energy. For example, gas mileage for vehicles has gone up over time, so that we are able to drive the same distance using less energy and producing fewer emissions. webDICE includes this effect by estimating the ‘energy intensity’ of the economy: how much output we can produce with a given amount of energy. We can estimate our current energy intensity from data and we know that energy intensity has declined over time. webDICE assumes that this trend continues, resulting in a gradual reduction in emissions per unit of output (although if economic activity grows faster than energy intensity goes down, emissions can overall increase).

While webDICE assumes that energy intensity goes down over time, the rate of decline slows down. That is, efficiency improvements get harder over time. You can set to rate of decline of energy intensity (or equivalently, the rate of efficiency improvements in our use of energy). If you set the value at zero, our energy intensity does not go down over time. We never improve our efficiency. As you increase the value, we improve our efficiency more quickly and we become less energy intensive. The following figure illustrates how carbon intensity evolves for choice of this parameter:

Cumulative industrial emissions are limited by the total available amount of fossil fuels. This cap is currently set at 6,000 gigatons of carbon dioxide, but users can increase this amount. more

### Eta

Eta is equal to the elasticity of margin utility.

### Fossil fuel limit

The fossil fuel limit is the total amount of fossil fuels that can be used. Cumulative industrial emissions have to be under the fossil fuel limit. The limit is currently set at 6,000 gigatons of carbon dioxide, but users can increase this amount.

### Harms

The harm parameter determines the climate sensitivity.

### Labor

The economy in webDICE uses labor and capital to produce output. The labor supply is assumed to grow with the population. webDICE starts with the current population (just over 6 billion people) and assumes that it will grow to a level that you can set. The default level is 8.6 billion. You can adjust this between 8.0 and 12.0 billion. The following figure illustrates how the growth in population changes for various parameter choices:

more

### Linear carbon model

Recent studies estimate that peak warming is linearly proportional to cumulative carbon emissions [8]. The linear carbon model uses this relationship to estimate temperature change as a function of cumulative emissions. The model does not include a carbon cycle or equations for radiative forcing, and as a result values for total carbon in the atmosphere or the upper and lower oceans are not computed. The default is set so that temperatures increase when cumulative emissions are 1 trillion tons C. This corresponds roughly to the default climate sensitivity of 3.2°C, which is the default value. Because the linear carbon cycle implicitly includes both a carbon cycle and radiative forcing, you cannot separately set the climate

[8] Allen, M. R. et al (2009) Warming caused by cumulative carbon emissions towards the trillionth tonne, Nature, 458:1163-1166.

### Lower ocean temperature increase

The lower ocean temperature increase is the change in temperature from pre-industrial levels of the lower ocean.

### Objective function

The objective function in the model is the discounted sum of utility:

$W=\sum_{t=1}^{Tmax}U[c(t),L(t)]R(t),$

where $U$ is utility, $L(t)$ is the population in period $t$, and $R(t)$ is the discount factor determined by the pure rate of time preference:

$R(t)=\frac{1}{\left(1+\rho\right)^{t}}.$

### Optimization mode

In optimization mode, the computer finds the path of emissions reductions that maximizes the objective function. It then translates this into a carbon tax that achieves those reductions. The optimal tax is determined by the assumptions you choose in the model. It is also affected by two parameters that do not otherwise affect model output: the pure rate of time preference and the elasticity of marginal utility of consumption. These parameters combine to determine the discount rate used by the optimizer to determine how to value future consumption.

#### Installation

Optimization in webDICE depends on the IPOPT library which interfaces with our python codebase through the pyipopt package. IPOPT is capable of using several linear solvers — webDICE uses the MA57 solver from HSL. IPOPT also depends on the LAPACK and BLAS libraries. This of course means that the user will have to install several libraries in order for the optimization to run successfully. General steps for installation on a Linux or Macintosh system are as follows (we have unfortunately not yet tested these steps in a Windows environment).

If you're able to obtain a license for the HSL libraries, download the source code. If you're unable to obtain an academic license for the solvers, there is a basic set publicly available that includes the MA27 solver, which should work fine. In either case, you'll need to accept a license before you can download the code. A basic configure, make, make install should suffice to install these libraries.

Download the source code for IPOPT, and uncompress it. Change into the ThirdParty/Metis directory, and execute the get.Metis file. This will download the source code for Metis, which will be installed during the IPOPT installation.

LAPACK and BLAS libraries are already installed in OS X, in the VecLib Framework, so you don't need to install these. On a linux box, you should change into the ThirdParty/Lapack and ThirdParty/Blas directories, and execute the get.Lapack and get.Blas files. As with Metis, source code will be downloaded which will install during the installation of IPOPT.

IPOPT installation, like HSL, should be a straight-forward configure, make, make install.

An updated pyipopt is not available through pip or easy_install. You should download the source code, or clone the repo from footnote [^2], move into the directory, and python setup.py install. However, for installation to complete successfully, it will likely be necessary to modify the setup.py file. Adjust the IPOPT_DIR variable based on the values you used in configuring IPOPT (e.g./usr/local). Eg, we pass --prefix=/usr/local to our configure scripts, and so leave this as is. Depending on the solver you're using and other environmental settings, one may need to adjust the extra_link_args, library_dirs, and libraries lists. If you have followed the steps above, you should only need to comment out 'coinmumps', and uncomment 'coinhsl' in the libraries list.

It is possible to install and run webDICE without installing any of the optimization libraries. Executing an optimized loop is of course not possible under such a scenario, and will result in a runtime exception. However running other portions of webDICE should succeed without trouble.

#### Execution

Efforts have been made to create an optimization routine that balances accuracy with the the swiftness that is required for a web-based application. However, calculating an optimized policy naturally takes longer than other policy choices. Currently, a user can expect this to take between 10 and 20 seconds.

Some features of webDICE increase the solve time of the optimization routine. For the time being, the added computation of the BEAM carbon model increases the necessary solve time to a point which is infeasible for web delivery. As such, optimizing a scenario that includes the BEAM model has been disabled.

### Optimized policy

In optimization mode, webDICE weighs the costs and benefits of reducing emissions and finds the emissions reductions that balance the two. That is, it finds emissions reductions that produce the overall best economic performance given the costs of climate change and the costs of reducing emissions. When performing these calculations, the model uses utility people get from output rather output or GDP. For example, when comparing spending resources today to reducing emisisons — making people alive today worse off — to reduce harms from climate change in the future — making people in the future better off — the model compares the utility of the affected people. See utility. The model also can discount the utility of future people based on a discount rate that you choose.

The choices you make elsewhere, regarding climate sensitivity, harms from climate change, technological growth, abatement costs, and so on will affect how much we should reduce emissions. For example, if you choose a pessimistic scenario, such as high climate sensitivity or very bad harms from climate change, it will be desirable to reduce emissions more. Conversely, if you choose an optimistic scenario, we will not want to reduce emissions as much. Optimization mode allows you to see how your choices affect climate change policy. more

### Parameters

Parameters are the inputs into the model. They include values determined from current data, such as the population, the amount of capital, current temperatures, emissions, and so forth, as well as inputs that determine the future value of these items, such as the maximum population, the decline in energy intensity, and the pure rate of time preference.

### Participation

The participation fraction is the percentage of global emissions that are subject to the user-selected climate treaty. Setting the parameter less than 1 means that not all countries participate in an emissions control regime or some industries are exempt, or a combination of both. For example, China is about 20% of global emissions. If you set the participation to 0.8, this would produce a result roughly equivalent to excluding China. The result is only rough, however, because the model does not have separate countries or industries.

The participation fraction does not affect the emissions control level required by the treaty. Instead, it determines what portion of the world has to meet that level of emissions reduction. As participation goes down, the costs of meeting a target reduction goes up because reductions are concentrated on a smaller part of the economy. The abatement cost parameter (in the technology tab) determines how fast these costs go up. more

### Population

See labor supply.

### Productivity

Total Factor Productivity (TFP) describes how efficiently capital and labor produce economic output. The initial value of this parameter is set to produce 2005 output given observed capital and labor in that year. webDICE assumes that TFP will continue to increase in the future, but the rate at which it increases will slow down, like a hill that is steep at the bottom and gets shallower at the top.

The default rate at which TFP growth declines is 1%. You can adjust this parameter between 0.5% and 1.5%. Higher values means growth slows down more quickly. The following figure illustrates how changing the rate at which TFP growth declines affects the evolution of productivity (the parameter is labeled $\Delta_{A}$)

more

### Pure rate of time preference

The pure rate of time preference, represented by the Greek letter rho, $\rho$, determines how we count people living at different times when estimating the effects of climate change and policies to reduce climate change. Suppose that we can spend USD1 now to reduce the effects of climate change and that doing so produces a USD2 benefit in the future. To determine whether this expenditure is wise, we have to be able to compare the cost of USD1 to current people with the USD2 benefit to future people. The starting point in the model is to calculate the change in utility from the losses and gains. If the current people are wealthier, for example, they may not gain from an extra USD2 as much utility as current people would lose by spending USD1 to reduce emissions. After translating the dollar gains and losses into utility, we need to be able to compare the utility of people living in different time periods which requires a value for the pure rate of time preference.

The model generally adds up the changes in utility to determine the overall change in welfare from a chosen policy, but reduces the change in utility of people living in the future by a discount factor. The discount factor is calculated using the pure rate of time preference in the same way we would calculate the present value of money when we know the interest rate. If we let the pure rate of time preference equal $\rho$, the utility of people living in time period t is $U_{t}/(1+\rho)^{t}$. The higher the pure rate of time preference, the more future changes in utility are discounted.

The use of a positive pure rate of time preference is controversial. Some argue that it is unethical to count people living in the future less than we count current people. Others argue that a positive pure rate of time preference reflects how people actually behave and that a model of climate change cannot produce realistic scenarios and policies unless they reflect actual behavior. We take no stance on the value for the pure rate of time preference. You can choose to set it from anywhere between 0 and 10%. The default, which simply reflect the default used in the DICE model generally and not necessarily our views, is 1.5%.

Radiative forcing determines how much the CO2 in the atmosphere increases temperatures. It is a measure of the difference between incoming energy from the sun and the energy radiated back out to space from the earth. Any differ results in temperatures increasing. You can choose how much a given amount of CO2 in the atmosphere increases temperatures by choosing the value of the climate sensitivity. The default value is 3.2, which means that a doubling of atmospheric CO2 produces a 3.2°C temperature increase. more

### Savings

webDICE assumes that people consume a portion of output and save the rest. The default savings rate is 22% of output. Some versions of DICE solve for the optimal savings rate. webDICE does not support optimization of the savings rate because the optimal savings rate is relatively insensitive to assumptions and optimization would significantly slow down the model. Instead of optimization, webDICE allows users to choose a savings rate between 15% and 25%.

### Simulated carbon tax

You have the option of imposing a carbon tax. You can choose the rate for three different years: 2050, 2100, and 2150. The model sets the rates between those years by interpolating. For example, if you set a rate in 2050 of USD100/ton of CO2 the model gradually increases the rate between now and 2050 to reach that level. If you set the rate in 2100 to USD150/ton of CO2 the rate gradually increases from 2050 to 2100.

The tax rate is capped at the level that is needed to eliminate fossil fuels. For example, if a tax rate of USD200/ton CO2 would be enough to cause a complete switch to wind and solar energy, the model assumes that this is the maximum rate, so setting the tax at USD250 ton CO2 produces no additional reductions.

### Social cost of carbon

The social cost of carbon (SCC) is a commonly cited metric which estimates the increase in harms from additional emissions. To estimate this, webDICE runs the model once (under your chosen parameters) and then again with one additional ton of CO2 and calculates the difference in the two cases. The increase in harms from the additional ton is the social cost of carbon. U.S. government agencies are required to use this metric in cost-benefit evaluation of new programs. Although the commonly used term is social cost of carbon, it is standard to report the value in dollars per ton of carbon dioxide, not carbon. webDICE follows this practice.

webDICE calculates the social cost of carbon for each period by sequentially adding the additional ton in each period. For example, the social cost of carbon in 2055 is the increase in harms one an additional ton of CO2 is added in the year 2055. more

See carbon tax

### TFP

TFP stands for total factor productivity. See productivity

### Time steps

The model uses 10 year time steps. It shows output for 20 time steps or 200 years, starting in 2005. It runs for runs for 60 periods (600 years) in the background so that the end period does not influence the calculations of the social cost of carbon or the optimal emissions control rate. You can see all 60 periods by downloading the CSV file.

### Total carbon emitted

Total carbon emitted is the sum of emissions in all prior years.

### Total consumption

Utility discounted in a given period is the utility in that period discounted to its current value using the pure rate of time preference.

### Utility

Utility is a measure of our well-being, of how well off we are. Although well-being depends on many things, webDICE treats utility as a function of consumption. It assumes that as we get richer we are better off. It assumes, however, that the benefits of getting richer still are smaller. This is sometimes called the ‘declining marginal utility of income’. It means that going from, say, poor to middle class produces great benefits, from middle class to wealthy also makes you better off but the gains are smaller. Going from wealthy to incredibly rich produces smaller gains still. The intuition is simply that a dollar matters less to Mark Zuckerberg thanks to a poor person.

You can choose the rate at which the gains from getting richer goes down by choosing the variable called ‘eta’ or $\eta$. If you change eta, the model runs and almost all of the output will unaffected because utility does not enter into most of the calculations. The one place where changing eta will change the results is in optimization. If you increase eta so that the gains from getting richer are smaller, the optimizer will put less weight on the wealthy and more on the poor. The reason is that the optimization focuses on utility. As eta gets larger, giving another dollar to a wealthy person produces a smaller increase in their utility.

One important but technical problem with changing eta is that the graphs of utility with different etas are not comparable. more

### Warning screen

Some choices of parameters can produce output that is physically impossible. For example, if you choose a damage function that produces very large harms, the output of the economy may not be large enough to support the population, even at subsistence levels. Because the population level in the model is specified externally, the population remains high even though they have nothing to eat. Or if you choose a climate treaty that demands large reductions in emissions yet limits the portion of the world that is subject to the treaty, the reductions may be impossible to achieve. For example, if a treaty requires a 30% reduction in emissions, and the nations subject to the treaty only emit 20% of the total, there is no way to achieve the required reductions.

The model flashes a warning label when the output is physically impossible. While physically impossible, the results are meaningful because they show that under given scenario, something must change. For example, if the economy cannot support the population, the population has to go down. Seeing the results in these cases, therefore, may be helpful in understanding the possible effects of climate change. After clicking on the warning label, you can see the results from that scenario.

### webDICE

webDICE is a web-based version of the most widely-used model of the economics of climate change, DICE (Dynamic Integrated model of Climate and the Economy). webDICE allows users to see the possible effects of climate change using DICE, and also to understand how changes to the model’s assumptions change those effects. DICE was developed by Professor William Nordhaus. His book, A Question of Balance, Weighing the Options on Global Warming Policies, is a good place to find more details. Nordhaus’s equations, code and additional documentation are available (at http://nordhaus.econ.yale.edu/DICE2007.htm. Due to computational considerations, we have re-coded the model into a different computer language, python, so that this version may not run identically to the version in A Question of Balance or to other versions found on Professor Nordhaus’s website.

webDICE is based on a model of the global economy in which industrial activity produces emissions, which cause climate change, which in turn harms the economy. The basic model runs under the assumption that there is no policy in place to control emissions. The parameters in the model are based on available data and, where there is no data, a guess. Many these guesses are just that — guesses — and you should view the outcome as just one possible scenario. Clicking on the ‘run model’ button shows the results under this scenario. You can change the assumptions to see how these changes affect the economy and people in the future. For example, you can change how fast the price of clean energy goes down (or if it will), how fast the economy will grow, the extent of the harms from climate change, and most of the other assumptions of the model to see what your views on these matters mean for the future.