Specification and Global Extension of the RICE Model

From a mathematical perspective, the RICE model is an optimization algo-rithm, whose objective function is the maximization of the prevailing social welfare of the region considered, and its limiting factors are the equations describing economic and natural relationships. Since the model is intended to capture very complex interrelationships with many components that are difficult to define mathematically, simplifications and assumptions have to be applied. These will be discussed at a later point, together with the specifi-cation of the model. The equations of the model can be divided into four groups: relationships concerning the objective function, the economic rela-tions and the impacts of climate change, as well as the optimum in the market of fossil fuels. This section describes the model’s equations in this order.

Objective Function

The first problem that arises with regard to the objective function is the def-inition of social welfare. One part of the problem is a static measurement

208 W.D. Nordhaus (1994): Managing the Global Commons: The Economics of the Greenhouse Effect, Cambridge, MA, The MIT Press

difficulty regarding how welfare can be quantified, while the other part—

which derives from the fact that the model examines long time spans and several periods—concerns the question of how the conversion of values be-tween periods can be managed, i.e. how the model can be dynamized. The RICE model addresses the measurement-related difficulty of the first prob-lem using the concept of ‘generalized consumption’, which is commonly used in macroeconomics. This means that the model assumes that the opera-tors of the economy can produce and consume a single type of goods; thus, investments can also be used to increase the production of the good con-cerned. In this way, the changes concerning this single good provide an indi-cation of the changes in social welfare.

From an economic perspective, the question of the trade-off between differ-ent periods is one of the key problems of the efforts aimed at reducing climate change. The main dilemma refers to the number of goods that society should renounce in order to ensure that the production potential of future generations is preserved. Each generation in each period has to decide what proportion of the goods meant to be invested is used to increase productivity and how much is used to mitigate climate change. The first group of goods produces a larger amount of consumable goods but greater climate risks in the next period, while the latter leaves an unchanged amount of goods and lower cli-mate risks for future generations. The RICE model includes a so-called dis-count factor for addressing the problem of intertemporality, which represents the society’s willingness to make trade-offs between periods. On this basis, the objective function of each region considered can be described with the following formula:

𝑊_𝐽 = ∑ 𝑈[𝑐_𝐽(𝑡), 𝐿_𝐽(𝑡)]𝑅(𝑡)

𝑡

𝑊_𝐽 stands for the social welfare of region 𝐽, which needs to be maximized during the optimization process. 𝑐_𝐽(𝑡) represents the per capita consumption of region 𝐽 during period 𝑡, while 𝐿_𝐽(𝑡) is the size of its population.

𝑈[𝑐_𝐽(𝑡), 𝐿_𝐽(𝑡)] is the utility of consumption, while 𝑅(𝑡) stands for the dis-count factor, which is the same for all regions. The model aims to maximize the sum total of discounted utilities over all the periods of the entire time span considered, in a regional breakdown. With regard to the utility function, the RICE model assumes that higher levels of consumption generate greater utility and that there is the diminishing marginal utility of consumption as consumption increases. The utility function of consumption—also known as the economic value of welfare—is obtained by multiplying the size of the

population of the given period [𝐿_𝐽(𝑡)] by the utility of per capita consump-tion, which can be interpreted as follows:

𝑈[𝑐_𝐽(𝑡), 𝐿_𝐽(𝑡)] = 𝐿_𝐽(𝑡)𝑐_𝐽(𝑡)^1−𝛼− 1 1 − 𝛼

In this equation, parameter 𝛼 is the elasticity of the marginal utility of con-sumption, i.e. the selection of this parameter determines the curvature of the utility function. The empirical DICE and RICE models assume a limit value of 𝛼 = 1, which yields the following utility function, the so-called Bernoul-lian utility function:

𝑈[𝑐_𝐽(𝑡), 𝐿_𝐽(𝑡)] = 𝐿_𝐽(𝑡){𝑙𝑜𝑔[𝑐_𝐽(𝑡)]}

The key parameter of the utility function is the population, whose future evo-lution is calculated by the model on the basis of the initial population of the region [𝐿_𝐽(0)] and the rate of population growth [𝑔_𝐽^𝑝𝑜𝑝(𝑡)] as follows:

𝐿_𝐽(𝑡) = 𝐿_𝐽(0)𝑒𝑥𝑝 (∫ 𝑔_𝐽^𝑝𝑜𝑝(𝑡)

𝑡 0

)

When the RICE-99 model was elaborated, population forecasts predicted a declining growth rate, therefore, choosing it as a constant proved to be inef-fective. For this reason, the model calculates the growth rate on the basis of an initial growth rate [𝑔_𝐽^𝑝𝑜𝑝(0)] and a constant rate of decline [−𝛿_𝑗^𝑝𝑜𝑝], which varies from one region to another, as follows:

𝑔_𝐽^𝑝𝑜𝑝(𝑡) = 𝑔_𝐽^𝑝𝑜𝑝(0)𝑒𝑥𝑝(−𝛿_𝑗^𝑝𝑜𝑝𝑡)

In the empirical model, the initial growth rate was 1.5% and the maximum global population, also known as the population limit of the given period, was 11.5 billion.

According to the assumption related to the discount rate, taking the same population size and per capita consumption, the marginal social utility of consumption is higher. The original empirical model operates in periods of ten years, in which flow variables are reported as flows per year, while stock variables are measured at the end of the period. The discount rate depends on the pure time preference of society [𝜌(𝑣)] as follows:

𝑅(𝑡) = ∏[1 + 𝜌(𝑣)]⁻¹⁰

𝑡

𝑣=0

Due to the long time period considered, the selection of the pure time prefer-ence (or discount rate) has a significant impact on the results of the model.

In the first DICE model, Nordhaus used a constant discount rate of three per-cent, but later this was strongly contested by other researchers. Many re-search results on the human perception of time show that the social discount rate declines as the time span increases. Accordingly, one of the improve-ments made to the model was the dynamization of the discount rate with a constant rate of decline [−𝑔^𝜌𝑡]; thus the corrected version uses the following equation:

𝜌(𝑡) = 𝜌(0)exp⁡(−𝑔^𝜌𝑡)

In the empirical model of 1999, the discount rate [𝜌(0)] starts at 3 percent in the initial year of 1995 and declines to 2.3 percent in 2100 and to 1.8 percent in 2200.

Economic Relationships

The RICE model calculates the economic output of the region using the Cobb-Douglas production function, which is the most commonly used func-tion in tradifunc-tional growth models. One of the innovafunc-tions of the model is that, in its production function, the output [𝑄(𝑡)]—in addition to the capital stock [𝐾(𝑡)], labour [𝐿(𝑡)] and technology [𝐴(𝑡)], which are traditionally taken into account—is also affected by energy services from the combustion of fossil fuels [𝐸𝑆(𝑡)], as well as their unit costs [𝑐_𝐽^𝐸(𝑡)]. Another innovation of this model is the integration of a so-called ‘damage coefficient’ [Ω_𝐽(𝑡)] into the production function, which represents the impact of climate change on economic output. The production function can, therefore, be formulated as follows:

𝑄_𝐽(𝑡) = Ω_𝐽(𝑡){𝐴_𝐽(𝑡)𝐾_𝐽(𝑡)^𝛾𝐿_𝐽(𝑡)^{1−𝛽𝐽−𝛾}𝐸𝑆_𝐽(𝑡)^𝛽𝐽− 𝑐_𝐽^𝐸(𝑡)𝐸𝑆_𝐽(𝑡)}

The production function has three parameters that can be calibrated: 𝛾 is the elasticity of output with respect to changes in capital and 𝛽_𝐽 is the elasticity of output with respect to changes in energy services. In this model, techno-logical progress is reflected in two different ways: in productivity growth and in the improvement of energy conversion efficiency. 𝐴_𝐽(𝑡) indicates the ex-ogenously increasing productivity, whose future evolution—similarly to population—is influenced by a region-specific and time-varying parameter reflecting the growth rate [𝑔_𝑗^𝐴(𝑡)] as follows:

𝐴_𝐽(𝑡) = 𝐴_𝐽(0)⁡𝑒𝑥𝑝 (∫ 𝑔_𝐽^𝐴(𝑡)

𝑡 0

)

There is significant uncertainty concerning the projection of the future evo-lution of productivity. The RICE model assumes that the improvement of productivity will slow down. The growth rate for the period considered can be calculated on the basis of its initial value [𝑔_𝑗^𝐴(0)] and a constant rate of increase [−𝛿_𝑗^𝐴𝑡] as follows:

𝑔_𝑗^𝐴(𝑡) = 𝑔_𝑗^𝐴(0)𝑒𝑥𝑝(−𝛿_𝑗^𝐴𝑡)

Another type of technological development is related to the improvement of energy conversion efficiency. The energy service [𝐸𝑆(𝑡)] included in the production function is calculated by multiplying the amount of fossil fuels used [𝐸_𝐽(𝑡)] by their conversion factor [𝜍_𝐽(𝑡)].

𝐸𝑆_𝐽(𝑡) = 𝜍_𝐽(𝑡)𝐸_𝐽(𝑡)

In this model, the growth of energy generating technologies that do not pro-duce carbon dioxide emissions is taken into account through the increase of the conversion factor. Similarly to the population and productivity, the con-version factor is predicted with the following equation including a time-var-ying growth rate parameter [𝑔_𝐽^𝑍(𝑡)] and a constant rate of increase [−𝛿_𝑗^𝑍𝑡]:

𝜍_𝐽(𝑡) = 𝜍_𝐽(0)⁡𝑒𝑥𝑝 (∫ 𝑔_𝐽^𝑍(𝑡)

𝑡 0

)

𝑔_𝐽^𝑍(𝑡) = 𝑔_𝐽^𝑍(0)⁡𝑒𝑥𝑝(−𝛿_𝑗^𝑍𝑡)

The main difference between the DICE and RICE models is that while the DICE model considers the entire world economy and its emissions to be a single agent, in an aggregate manner, in the RICE model, regions can trade their carbon dioxide emission allowances. In growth models that assume a traditionally closed economy—due to the generalized consumption con-cept—the region considered uses one part of the economic output (or the GDP produced) [𝑄_𝐽(𝑡)] for consumption [𝐶_𝐽(𝑡)] and the other part for in-vestment [𝐼_𝐽(𝑡)]. In the RICE model, potential revenues and costs deriving from the trading of carbon dioxide emission allowances are added to the available emission allowances. The model does not examine other types of commercial activities. Therefore, the budgetary constraint of the region con-cerned is as follows:

𝑄_𝐽(𝑡) + 𝜏_𝐽(𝑡)[Π_𝐽(𝑡) − 𝐸_𝐽(𝑡)] = 𝐶_𝐽(𝑡) + 𝐼_𝐽(𝑡)

Π_𝐽(𝑡) is the number of allowances allocated to the region and 𝜏_𝐽(𝑡) is the unit cost of carbon dioxide emissions. It is important to note that the RICE model primarily regards energy carriers as fossil fuels, and thus provides the value of the parameter 𝐸_𝐽(𝑡) on the basis of its carbon content, which means that the fuel used is equal to its carbon content. Energy use, therefore, reduces the amount of carbon dioxide emissions available to a region. As in this model fossil fuels are represented in terms of both quantity and price by a parameter each; during the empirical implementation, the estimation of such parameters should be done by the weighted aggregation of different technologies.

Depending on whether the region considered uses more or less carbon than it is entitled to, its budgetary constraint may decrease or increase. The 𝜏_𝐽(𝑡) unit cost parameter can be interpreted both as the price of the carbon dioxide emission quota and as the Pigovian tax payable for carbon dioxide emissions. The RICE model assumes that emission trading between regions is efficient, thus the price of carbon dioxide emissions is equalized, and its value is at least zero.

The region’s total consumption can be calculated by multiplying per capita consumption by the population size:

𝐶_𝐽(𝑡) = 𝑐_𝐽(𝑡)𝐿_𝐽(𝑡)

In order to determine the amount of capital available to the region considered during a given period, the capital stock of the previous period must be re-duced by a depreciation rate 𝛿_𝐾. The empirical RICE model assumes that the capital depreciates at 10 percent per year, and since investment is measured at annual rates, the depreciation should be adjusted to the ten-year period as follows:

𝐾_𝐽(𝑡) = 𝐾_𝐽(𝑡 − 1)(1 − 𝛿_𝐾)¹⁰+ 10 × (𝑡 − 1)

The production function includes the cost per unit of the region’s energy ser-vice [𝑐_𝐽^𝐸(𝑡)], which is the sum total of the world market price of energy [𝑞(𝑡)], which is assumed to be equalized globally, and a markup [𝑚𝑎𝑟𝑘𝑢𝑝_𝐽^𝐸(𝑡)], which is different in each region. In the RICE model, the markup captures the costs of energy transportation and distribution, the amount of which is assumed to be constant over time. The improvement of transportation and distribution technologies is represented by the increase of the previously discussed productivity.

𝑐_𝐽^𝐸(𝑡) = 𝑞(𝑡) + 𝑚𝑎𝑟𝑘𝑢𝑝_𝐽^𝐸(𝑡)

The first DICE and RICE models used fixed energy costs on the basis of the explicit assumption that the available stocks are inexhaustible. However, this was strongly criticized in the related literature, as the stocks of fossil fuels—

which are at the center of the model—are limited, and due to the considered time period of two hundred years, the depletion of the stocks may have a significant impact on world market prices. In the improved models, world market prices for energy are determined on the basis of each region’s energy demand. The cumulative energy demand [𝐶𝑢𝑚𝐶(𝑡)] related to global eco-nomic output can be calculated as the sum of the demand registered at the end of the previous period and the annual demands adjusted with the 10-year period:

𝐶𝑢𝑚𝐶(𝑡) = 𝐶𝑢𝑚𝐶(𝑡 − 1) + 10 × ∑ 𝐸_𝐽(𝑡)

𝐽

As the RICE model focuses on fossil fuels, which are available to a limited extent, the model expects that, in the future, at a certain amount of global demand [𝐶𝑢𝑚𝐶^∗], there will be an inflection point in the extraction costs of energy carriers, beyond which their marginal costs will begin to rise. Accord-ingly, the world market price for energy is calculated with the following equation:

𝑞(𝑡) = 𝜉₁ + 𝜉₂[𝐶𝑢𝑚𝐶(𝑡) 𝐶𝑢𝑚𝐶^∗ ]

𝜉₃

𝜉₁, 𝜉₂ and 𝜉₃ are fit parameters, whose modification allows for the energy supply curve to be estimated on the basis of empirical observations and ex-pectations. Due to the overall growth framework of the model, operators ef-ficiently allocate their use of energy sources, thus the empirical model con-firms the intuition, and low fossil fuel prices increase over time.