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A coupling algorithm for computing large-scale dynamic computable general equilibrium models
Economic Modelling 16 Ž1999. 455]473
A coupling algorithm for computing large-scale dynamic computable general equilibrium models Zili YangU Department of Energy, En¨ ironmental and Mineral Economics, The Pennsyl¨ ania State Uni¨ ersity, 221 Walker Bldg., Uni¨ ersity Park, PA 16802, USA
Abstract Large-scale dynamic computable general equilibrium ŽCGE. models are widely used for policy evaluation and economic forecasting in the studies of global environmental problems, such as climate change. Investigating these issues requires that CGE models have long time spans. Widely used dynamic mechanisms in CGE modeling, such as myopic expectations and calibration on ‘balanced growth’ path, are inadequate for such a long time scale. To improve the performances of long-term CGE models, a coupled algorithm that combines CGE modeling with optimal growth modeling approach, is introduced in this paper. The caveats of existing methods and improvements from the coupled algorithm are explored analytically as well as exemplified in the EPPA model, one large-scale dynamic CGE model with long time horizon. Numerical results from the EPPA model show the superiority of the coupled algorithm over existing treatments of dynamics in CGE models. Q 1999 Elsevier Science B.V. All rights reserved. JEL classifications: C61; C63; C68 Keywords: CGE modeling; Optimal growth; Modeling methods; Economic forecasting
1. Introduction Computable general equilibrium ŽCGE. models are widely used for empirical economic analysis and policy evaluations. Thanks to the growing capacity of U
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computing technologies, CGE models have evolved from elegant stylized small models wfor example: Meade’s model on international trade ŽMeade, 1955. and Harberger’s model on tax incidence ŽHargerber, 1962.x for highlighting economic theories to large, multi-sector, multi-region models that simulate economic relationships in great detail. CGE modeling also has become an important field of research. The basic methodology of CGE modeling as well as a plethora of CGE models developed in the 1970s and 1980s are brilliantly summarized by Shoven and Whalley Ž1992. as well as Scarf and Shoven Ž1984.. Most CGE models are static models built on a given data set. Economic analyses are based on comparisons between a baseline scenario and ‘counter factual’ scenarios induced by endogenous or exogenous shocks. Due to the richness of sectoral andror regional breakdown, CGE models are good analytical tools for trade policy and tax policy analysis. Such issues frequently involve the policy impacts on the welfare of different sectors, population groups and regions. The CGE modeling approach receives its fair share of criticism, especially from econometricians wfor example, see Lau Ž1984.x. The major criticism is focused on the parametric specifications of CGE models, which lack adequate statistical justification. Thus the results from CGE models are potentially compromised by their sensitivity to parameter assumptions. Despite those criticisms, CGE models remain effective analytical tools for indicative purposes. Very few large-scale CGE models are dynamic because of their intrinsic complexity, which will be outlined in the next section. Nonetheless, attempts have been made to build such models. Increasing concern about long-term impacts of some global environmental problems, such as greenhouse gas ŽGHG. emissions, have prompted the building of large-scale dynamic CGE models. Those models usually serve at least two purposes: they forecast long-run economic growth and the evolution of energy use patterns; they also support analysis of costs of carbon control policies, based on their richness of sectoral and regional breakdown. An early example of such models is OECD’s GREEN model ŽOECD, 1992.. More recently, several large-scale dynamic CGE models have been built for studying global environmental issues. They include MIT’s EPPA model ŽYang et al., 1996. which uses the GREEN data set, and the Australian Bureau of Agriculture and Resource Economy’s MEGABARE model ŽABARE, 1996., and the CETM model ŽMontgomery et al., 1997.. This list is not complete and more models are coming. The dynamic mechanisms of these models are different. In principle, they are built on either on myopic expectation or calibrated on some ad hoc paths. In those models, the most important intertemporal economic relationship } namely, saving, investment and capital formation } lacks any forward-looking mechanism. Instead, strong, exogenous assumptions are imposed on the intertemporal growth path. As a result, the forecasting capability of these dynamic CGE models is limited. In this paper, I propose a new algorithm to overcome the above shortcomings in long-run dynamic CGE models. The fundamental structure of the algorithm involves the coupling of a static CGE model with a matching Ramsey-type optimal growth model to replace the recursive dynamic mechanism in CGE models.
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Intertemporal optimal decisions on saving and investment are made in the optimal growth model, which feed into the static CGE model. This coupled CGE model has the essential desired properties of both a forward looking optimal growth model and a static CGE model. Namely, a time-consistent growth path to forecast future economy and a rich sectoral breakdown to conduct relevant policy analysis. Computationally, the coupled model is efficient. It has the same order of computing time with corresponding recursive dynamic model, if the initial searching condition is properly set. The remainder of this paper is organized as follows. In the next section, a brief summary and critique of recursive dynamic approach in CGE modeling and the rational for the coupling algorithm are presented; in Section 3, the new algorithm for coupling a static CGE model with a forward-looking optimal growth model is introduced; in Section 4, the algorithm is implemented into the EPPA model ŽYang et al. Ž1996.., a large-scale recursive dynamic model. A detailed comparison between old and new algorithms is also presented here. In the last section, the direction of further work is indicated.
2. Caveats in recursive-dynamic CGE models Most empirical CGE models are static models. One reason for this is the data requirements of dynamic CGE models. These are calibrated models, based on disaggregate input]output tables. It is virtually impossible to obtain input]output tables of given regions for consecutive periods as modelers desire. Another reason is that the static CGE models can fulfill many policy analysis tasks, using counterfactual simulations on a calibrated equilibrium to probe the impact of different price and quantity shocks. However, if a CGE model is designed not only for policy analysis but also for economic forecasting, where no data for calibration are available, the time dimension and certain mechanisms of intertemporal dynamics have to be included in the model. In practice, recursive mechanism is widely adopted by CGE modelers. A CGE model based on recursive dynamic principles is actually a series of static models solved sequentially. Because the calibration is only imposed on the initial period, the variables in subsequent periods in the reference path are calculated with changing exogenous and endogenous endowments. The outcomes from subsequent periods with no policy intervention can be treated as quantity Žendowment. shocks to the calibrated initial period. In other words, the reference scenario in a recursive dynamic CGE model is a series of ‘counter-factual’ equilibria based on the initial period. While endowment changes in subsequent periods can be defined as exogenous or endogenous, very strong assumptions about exogenous endowment changes, such as capital formation, may be required. Many important economic activities are intrinsically forward-looking. Current saving decisions of people are partially based on their expectations of future returns; future resource availability and expected policy shocks affect people’s current consumption patterns. The recursive approach in CGE models is not
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designed to deal with these issues. When such issues are encountered in recursivedynamic CGE models, special compromising assumptions are made to resolve the problems. In this paper, we direct our attention to saving and investment problem only and more complicated dynamic issues are indicated as future research direction, following the methodology provided in this paper. In CGE modeling practice, intertemporal decisions on saving and investment are based on conventional ad hoc assumptions. Two commonly adopted assumptions are either a ‘balanced growth path’ or myopic expectations. Under the ‘balanced growth path’ assumption, the ratios of the capital stock to the labor supply in future periods are the same as the ratio in the calibrated base period. If the growth of these two primary factors are disproportional in physical units, efficiency factors are applied to one or both primary inputs to maintain a constant ratio over time between the capital stock and the labor supply in efficiency units. The GREEN model ŽOECD, 1992. adopted this approach. Recently, there are some fully forward-looking CGE models that have been built on such a ‘balanced growth’ assumption ŽHarrison and Rutherford, 1997.. In these models, the economy is calibrated on a balanced growth path for the whole time horizon. Thus, the model generates an Arrow]Debreu economy in which commodities have both time and sector dimensions. The reactions of the economic system to the counter factual shocks are forward-looking. Alternatively, saving and investment decisions in a recursive CGE model can be based on certain myopic expectations. In each period, the saving and investment decision is determined by evaluation of lagged dual variables. Namely, Sa¨ Ž t . s In¨ Ž t . s f w r Ž t y 1 . rw Ž t y 1 . ,Y Ž t .x
Ž1.
Here, r is rate of returns to capital, w is wage rate and Y is the income of the representative consumer. This assumption makes the economy adjust its investment levels in each period according to the relative scarcity of capital in previous period. The EPPA model ŽYang et al., 1996. uses this assumption. Although the above two assumptions are less than desirable for describing saving and investment behavior in CGE models, they are still acceptable in many circumstances. Particularly, when the time horizon is short and the economy to be modeled is stable within the time horizon, the assumptions may be good approximations for the simulation of intertemporal dynamics. When the time horizon is short, say 5]10 years, the ratio of capital stock to labor does not change significantly, as long as there are no implausible and drastic changes in demographic composition, large-scale retirement of capital or sudden and very large increases in investment. For those economies that are in transition, or not stable, the balanced growth assumption is not proper even for a short time horizon, however. For example, economic reform in China resulted in a fast-growing economy and caused fundamental structural changes in the economy. The collapse of the Soviet political system also caused partial collapse of Russian economy. In both situations, the economy to be modeled is not on a balanced growth path. If the calibrated base
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period is followed by such events, the balanced growth path assumption can lead to large errors. In general, both recursive dynamic and ‘balanced growth’ path assumptions in CGE modeling are not consistent with the theories of economic growth. In effect, the notion of balanced growth is equivalent to the term ‘equilibrium’ in a static setting. One major premise of CGE modeling is that the calibrated base period represents an equilibrium state.1 In a recursive dynamic CGE model, all subsequent periods would be proportional ‘copies’ of the initial equilibrium except for exogenously imposed changes in policies andror resource availabilities. However, from a long-run perspective, the base period is no more than an arbitrary observation on growth path. It is implausible to assume that an economy, during an arbitrary initial period, is located on its long-run balanced growth path. The logic behind the assumption is contradictory to the economic theories explaining optimal economic growth patterns, such as turnpike theorems wsee Takayama Ž1985, ch. 7. for a summary of turnpike theoremsx. According to these theories, starting from an arbitrary base period, an optimal saving-investment plan can drive the economy to the long-run balanced growth path quickly. In general, the optimal capital labor ratio along the balanced growth path is different from that of the initial period. Therefore, the fundamental assumption on the calibrated equilibrium in the base period is not justified for the long-run growth path. The assumption of myopic expectations also has some severe problems. First, unless the economy is on the balanced growth path, myopic expectation is not consistent with fully foresighted expectations. Since the steady-state assumption is not true for any arbitrary initial period, the myopic expectation is not necessarily intertemporally optimal. Second, in counter-factual simulations, policy shocks always cause changes in relative prices. As a result, the price signals for the myopic expectations also change. However, these price change in current period are not consistent predictors for long-run capital formation. For example, if a policy shock in one period causes the relative price of capital to increase, that does not necessarily imply that long-run demand for capital will increase. Or, in the next period, a similar policy shock might cause the relative price of capital to decrease, yet it does not necessarily mean that capital formation should slow down. However, they would do so, according to Eq. Ž1.. In summary, the impact of current policy shocks on the long-run capital formation through the prism of myopic expectation is inconclusive, even arbitrary. Furthermore, the savings behavior of myopic expectations changes the disposable income of the representative agent. If a policy shock changes the size of the consumption ‘pie’ arbitrarily, the welfare analysis of the policy impact is questionable. Finally, the pure time preference of the representative consumer, which need not be specified in a model based on myopic expectations, does not affect his saving and investment decision. Therefore, saving and investment decisions are not consistent with optimal saving and investment decisions under a given pure time 1
In modeling practice, different types of ‘dis-equilibria’ can be built into the base period ad hoc.
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preference. As a result, the intertemporal welfare analysis of policy impacts lacks a clean foundation. These considerations lead to the following conclusions. While the assumptions of balanced growth path and myopic expectations are reasonable and acceptable for short-run dynamic CGE models, they are not satisfactory for long-run CGE models. If the long-run CGE model is to be used for both policy analysis and economic forecasting, a better treatment of long-run dynamics is required. The coupling algorithm, suggested in this paper, overcomes shortcomings of recursive dynamic or ‘balanced growth’ path assumptions in CGE modeling.
3. A coupling algorithm for long-run CGE models To achieve the dual tasks of policy analysis and economic forecasting, a long-run large-scale CGE model has to depart from the traditional assumptions that the initial period is on a balanced growth path. The assumption of equilibrium in the initial period should be used only for benchmark calibration purposes. The mechanisms of intertemporal saving and investment should be guided by the principles of optimal economic growth theories. According to these theories, if in the initial period the economy is not on its balanced growth path, it shifts to this path by adjusting saving and investment and thereby changing its laborrcapital ratio. To reflect this pattern, an efficient approach is to have an optimal growth model, with identical initial conditions and exogenous trend assumptions, interact with the CGE model, and provide the necessary saving and investment decisions for the CGE model. The coupling algorithm suggested in this paper implements this approach. To outline the coupling algorithm, we first express a recursive dynamic CGE model as a mapping of its primary factor inputs Žlabor and capital. to a state of economic equilibrium and Žusing M to indicate myopic. denote it as M w LŽ t ., K M Ž t .x. We assume that LŽ t . is exogenously specified and K M Ž t . is determined from certain recursive rules.2 Within the image of M Ž t ., are prices and quantities of commodities and factor demands. Some important aggregate variables, such as GNP and consumption, are extracted for interaction with the dynamic model. The corresponding optimal growth model, denoted as Dw LŽ t ., K D Ž t ., ID Ž t .x, is a dynamic system with the same exogenous input LŽ t ., the control variable ID Ž t . Žinvestment. and state variables wincluding K D Ž t .x. All state variables in DŽ t . have their counterpart in M Ž t .. The control variable ID Ž t . is interactive with M. M updates its investment based on ID Ž t .. In addition, DŽ t . has a free variable AŽ t . that can be adjusted independent of M. The analytical specification of AŽ t . can be interpreted as a Hicksian neutral technology progress factor. However, its function here is operational, rather than interpretational. As in other simple optimal growth 2 While labor supply, LŽ t ., is exogenous in most large-scale CGE models for various reasons, some models, for example MEGABARE ŽABARE, 1996., do have endogenous labor supply mechanism.
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models, AŽ t . is exogenous. We allow the adjustment of AŽ t . to ensure consistency of state variables between DŽ t . and M Ž t .. The procedure of the coupling algorithm is the following. First, the base period of D is calibrated to the initial condition of M such that YM w LŽ t 0 ., K M Ž t 0 .x s YD w LŽ t 0 ., K D Ž t 0 .x. Here Y is any common state variable in both M and D, GNP for example. Then D is solved for optimal saving and investment based on an initial AŽ t . path. The magnitude IDU Ž t 0 q 1. on the optimal solution path IDU Ž t . is selected to feed into the next period static CGE model M Ž t 0 q 1. as the aggregate saving-investment decision in the static model. Once IM Ž t 0 q 1. s IDU Ž t 0 q 1. is set, we solve the static CGE model for the period t 0 q 1, namely M Ž t 0 q 1.. The endogenous variables, such as GNP, from M Ž t 0 q 1. are then compared with the same variables from the optimal solution path of DU Ž t 0 . in its second period. If the difference between the two is smaller than certain pre-determined tolerance level, we can claim that the forward-looking saving-investment decision in DU Ž t 0 . is consistent with M Ž t 0 q 1.. Otherwise, the whole procedure is repeated from the first step by redefining AŽ t . according to certain tatonnement rules. When the convergence criterion regarding the endogenous variables is met, the algorithm moves to the next period to start all over again. A new optimal growth model starting from t 0 q 1 is defined with YM w LŽ t 0 q 1., K M Ž t 0 q 1.x s YD w LŽ t 0 q 1., K D Ž t 0 q 1.x as the initial condition and a new path of AŽ t . as the initial searching point. Fig. 1 is the schematic flow chart of the above-mentioned procedure. The procedure, with inference to the EPPA model, is illustrated in the flow chart in Appendix A. We claim that this coupling algorithm provides the correct aggregate saving and investment decisions for the CGE model to be coupled. First, M Ž t 0 . and DŽ t 0 . are started from the identical initial conditions for any t 0 in planning horizon of M. Therefore, the optimal intertemporal decisions in D are made upon the identical initial conditions and assumptions about exogenous endowment changes of M. Second, the forward-looking saving-investment decision IDU Ž t 0 q 1. in DŽ t 0 . generates the identical GNPs in period t 0 q 1 in DŽ t 0 . and in M Ž t 0 q 1.. In other words, the decision made in M Ž t 0 q 1. is consistent with the optimal forward-looking expectations from the previous period. On the balanced growth path or not, this decision is time-consistent and optimal regarding to the objective function in DŽ t .. Since we accept DŽ t ., a sequence of forward-looking optimal growth models starting at each period in M Ž t ., as the intertemporal decision maker for M Ž t ., a sequence of static CGE models, we in fact equip M Ž t . with a forward-looking mechanism. The combined outcome from the coupled algorithm has the strength of both CGE models and optimal growth models. The modeled economy is on an optimal growth path all the time for its time consistency. If the time horizon is sufficiently long, the ‘turnpike’ pattern is shown. On the other hand, sectoral demands for primary factors, impacts of policy shocks are still revealed by the static CGE model with rich sectoral and regional details. Also, the welfare analysis of policy impacts over time in the CGE model is consistent when using the pure time preference defined in the optimal growth model. Finally, we should point out that the coupling
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Fig. 1. Programming flow chart of coupled EPPA model Ž D is the dynamic model and M is the myopic submodel..
algorithm enhances forecasting capability of the CGE model to be coupled. Policy analysis tasks are still fulfilled by the CGE model.
4. Implementing the coupling algorithm in the EPPA model To highlight the necessity and efficacy of the coupling algorithm, we employ it in the EPPA model. The EPPA model is a global CGE model with 12 regions linked by bi-lateral trade matrix. In each region, there are eight production sectors and
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four consumer sectors. The model is driven by regionally-specified labor and capital supplies as well as sectoral-specified fixed factor supplies. The planning horizon is from 1985 to 2100 in 5-year intervals. The intertemporal dynamics of saving and investment decisions in the EPPA model now follows some myopic expectation rules similar to Eq. Ž1.. A detailed descriptions of the model is provided by Yang et al. Ž1996.. The EPPA model is an ideal candidate for testing the coupling algorithm because of its dual tasks of economic forecasting and policy analysis for a long time horizon. For executing the coupling algorithm, a matching aggregate Ramsey type optimal growth model is built. For the convenience, this optimal growth model is labeled DEPPA model. The DEPPA model is a optimal growth model with the same 12 regions as the EPPA model. Each region uses labor, capital and energy as inputs to produce an aggregate good called GDP. The objective function of the optimization problem is a social welfare function consisting of a weighted sum of intertemporal utility functions of 12 regions. Trade flows are allowed in the DEPPA model with the trade balances equivalent to interregional capital flows because only an aggregate homogenous good is produced here. The formal description of the DEPPA model is in Appendix A. Two underlining assumptions in the EPPA model make the setting up of the DEPPA model relatively easy. First, in the multi-layer CES production structure of the EPPA model, the current value of the substitution elasticity between labor and the bundle of capital, energy and fixed factor is 1.0. Thus the aggregate Cobb]Douglas production function in the DEPPA model is a good structural approximation of the EPPA model. Second, one of the closure rules in the EPPA model sets the balance on current account equal to zero Ži.e. each region is required to maintain trade balance .. The equivalent assumption in the DEPPA model is that capital flows between regions are zero. Therefore, the DEPPA model behaves like 12 ‘island’ economies without interactions among each other. The social welfare weights can be set arbitrarily in the DEPPA model. The assumption reduces the complexity of computation.3 The coupling algorithm is programmed in the GAMS language ŽBrook et al., 1992.. The EPPA model is solved by PATH solver ŽDirkse and Ferris, 1995. under MPSGE platform ŽRutherford, 1993. and the DEPPA model is solved by CONOPT solver ŽDrud, 1992.. The coupling algorithm is implemented by switching solvers and sharing information between intermediate iterations. The coupled model takes 3 h to solve on a Pentium Pro 200 computer with a fixed number Ž12. of tatonnement iterations between DEEPA and EPPA in each period. The original EPPA takes 10 min on the same machine. 3
In a detailed CGE model, the trade flows between regions can be enormous under the assumption of trade balance. We assume that the trade pattern does not have direct impact on domestic saving and investment behavior. Nevertheless, the more general assumption on capital flows, as defined in Appendix A, will be useful for many research and modeling purposes. We will indicate them later in the paper.
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For comparison purposes, four sets of simulations on the coupled model are conducted here. They are: Ži. Žii. Žiii. Živ.
a baseline scenario based on myopic expectations in saving decisions; a baseline scenario based on the coupled algorithm; a policy shock scenario based on myopic expectations in saving decisions; and the same policy shock scenario based on the coupled algorithm.
The policy adopted in Žiii. and Živ. is a version of so-called ‘AOSIS’ policy on carbon emission control. It requires the OECD region to curb its carbon emissions levels from 2010 onward at 80% of its 1990 level. This policy scenario is a good device for testing the properties of a global model linked by trade matrix, because of its draconian feature. To simplify the presentation and analysis, data and graphs are extracted from four regions out of total 12 regions in the EPPA model. To examine the convergence properties of the coupled algorithm, the maximum relative differences in GNP between the EPPA and DEPPA models are reported in Table 1. The definition of table entries is the following:
Max r
½
GNPM Ž r . y GNPD Ž r . GNPM Ž r .
5
Ž2.
where r is the index for regions. From Table 1 we can conclude that the two models are very close to each other after a few iterations. 4 Because the EPPA model is a complex non-linear system, convergence using the tatonnement method is not monotonic. The entries in Table 1 show the pattern. The rate of convergence can be improved by refinement of tatonnement method. Nevertheless, given the fact that the maximal differences in GNP in two models are smaller than 0.5% in most periods, the two models are consistent in the periods when making interactive saving and investment decisions. In addition, since investment is only a fraction of total GNP, the numerical errors reflected in Table 1 on saving and investment decisions are negligible. Fig. 2 plots the welfare changes of the coupled algorithm relative to that of recursive algorithm for USA, Japan ŽJPN., energy exporting countries ŽEEX. and China ŽCHN., respectively. In the main, welfare improvements are observed in most periods for all regions. Since the welfare indices are functions of consumption Žwhich is personal income net of saving., the welfare index is an indirect measure of savings. In addition, DEPPA’s maximand is a function of the intertemporal utility of regions, which in turn, are functions of consumption. Saving decisions affect welfare levels now and in the future. The saving decisions in recursive-dynamic CGE models are not based upon intertemporal welfare maximization. Therefore, it is unlikely that the recursive CGE model is on the optimal growth path. As the results, welfare indices from the recursive CGE model are inferior to 4
To test the convergence properties, a fixed number Ž12. of iterations are used instead of stopping by using a specified convergence criterion.
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Table 1 Maximum relative deviations of GNP between EPPA and DEPPA models Ž%. Iteration period
1
2
3
4
5
6
7
8
9
10
11
12
1990 1995 2000 2005 2010 2015 2020 2025 2030 2035 2040 2045 2050 2055 2060 2065 2070 2075 2080 2085 2090 2095 2100
14.26 13.25 12.54 10.25 7.44 5.19 4.01 4.22 4.52 4.82 5.09 5.29 5.42 6.01 6.68 7.13 7.28 7.16 6.82 6.32 5.79 5.26 5.05
13.65 10.34 8.49 6.81 4.57 2.98 1.96 2.15 2.30 2.43 2.52 2.84 3.38 3.90 4.34 4.66 4.82 4.80 4.64 4.38 4.06 3.77 3.51
11.87 7.31 4.52 3.34 1.70 1.06 1.36 1.73 2.10 2.38 2.63 2.88 3.03 3.09 3.06 2.93 2.76 2.52 2.49 2.45 2.37 2.32 2.22
10.23 4.91 1.74 0.91 0.98 2.03 2.23 2.58 2.96 3.23 3.40 3.66 3.86 3.95 3.98 3.90 3.77 3.58 3.35 3.07 2.80 2.50 2.25
8.79 3.19 0.66 0.92 0.80 1.38 1.33 1.38 1.44 1.46 1.46 1.45 1.45 1.53 1.61 1.66 1.70 1.74 1.77 1.74 1.71 1.66 1.58
7.54 2.03 0.51 1.01 0.73 0.64 0.43 0.51 0.80 1.11 1.36 1.52 1.60 1.58 1.50 1.32 1.11 0.86 0.66 0.62 0.58 0.53 0.46
6.45 1.27 0.42 0.69 0.43 0.73 1.04 1.40 1.77 2.06 2.40 2.63 2.75 2.79 2.76 2.58 2.39 2.13 1.86 1.52 1.29 0.98 0.77
5.52 1.07 0.30 0.28 0.33 0.81 0.94 1.10 1.25 1.34 1.38 1.42 1.50 1.54 1.59 1.58 1.56 1.50 1.44 1.29 1.16 1.03 0.88
4.71 0.90 0.23 0.22 0.22 0.36 0.24 0.17 0.23 0.39 0.56 0.67 0.69 0.66 0.56 0.39 0.25 0.21 0.22 0.27 0.33 0.40 0.43
4.02 0.76 0.18 0.12 0.09 0.16 0.38 0.67 0.99 1.33 1.61 1.79 1.88 1.87 1.80 1.61 1.41 1.14 0.91 0.59 0.43 0.25 0.15
3.42 0.65 0.14 0.11 0.09 0.37 0.54 0.74 0.92 1.05 1.18 1.26 1.32 1.35 1.37 1.30 1.22 1.13 0.98 0.78 0.67 0.50 0.40
2.91 0.55 0.10 0.07 0.11 0.28 0.29 0.28 0.22 0.15 0.13 0.17 0.16 0.15 0.18 0.23 0.26 0.28 0.35 0.40 0.34 0.37 0.31
those from the coupled model. In summary, welfare indices, ceteris paribus, are import criteria for evaluating the coupling algorithm. We should note that the changes in the second period for USA and CHN are large. The reason is that the initial saving rates is very high in China Ž30%. and is very low in USA Ž13%. in the EPPA model. In the second period the optimal saving rate from the DEPPA model is much lower for China and much higher for USA than in the base period. Such saving rates from the DEPPA alter consumption and cause drastic differences between the recursive EPPA results and forward-looking DEPPA results. The relatively low saving rate in the second period decreases the welfare levels in subsequent few periods in China. The relatively higher saving rate in the second period increases the welfare levels in almost all future periods in USA. Although a single period saving decision does not necessarily bear direct causality for all future welfare levels, we can still conclude that the saving rates from a given period need not be optimal for the long-run economic growth. Table 2 reports the present values of welfare indices discounted at 3% annually Žthe rate of pure time preference used in the DEPPA. from recursive and coupled algorithms, respectively. Method A is calculated by setting the base period at 1 and summing up all 24 periods. Method B provides the results from Method A net of the values of the
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Fig. 2. Welfare changes in reference scenario Žcoupled vs. recursive..
second period. The values of welfare indices from the coupled algorithm are consistently higher in Method B and are higher for three regions except USA in Method A. The lower present value of welfare index for USA in Method A is because of higher saving in the second period Žas shown in Fig. 2.. Nonetheless, the overall improvements on welfare from coupled algorithm over recursive algorithm are there, even for USA. The two methods of calculating present value of welfare indices show that coupled algorithm is better than recursive algorithm by so-called ‘catching up’ criterion Žsee Beavis and Dobbs, 1990.. Figs. 3]6 show welfare losses Žor gains. due to the AOSIS policy in four regions. USA and Japan face carbon emission constraints directly. As the results, demands for fossil fuels decline and welfare levels decrease. Energy exporting countries lose a major buyer ŽOECD. of their energy exporting commodities. They experience an even larger welfare loss because of the change in the terms of trade. In contrast, China has a very moderate welfare gain due to the change in the terms of trade under the AOSIS scenario. The results show that the recursive and coupled algorithms have different implications on the costs for carbon control policies. In the cases of the USA and
Table 2 The present value of welfare indices USA
JPN
EEX
CHN
Method A
Recursive Coupled
2.09700 2.05194
2.17133 2.18470
2.77094 2.80478
3.35900 3.42986
Method B
Recursive Coupled
1.58294 1.58669
1.65314 1.67170
2.02454 2.05619
2.49653 2.50270
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Fig. 3. Welfare changes: USA.
Energy Exporting Countries, the AOSIS policy incurs lower costs from the recursive algorithm than they are from the coupled algorithm. On the other hand, the AOSIS policy incurs higher costs from the recursive algorithm than they are from the coupled algorithm in Japan. In case of China, the gains from the AOSIS policy is lower from the recursive algorithm. The magnitude of the differences in the policy generated welfare changes between the two algorithms are significant. Since the only structural difference between the two algorithms is the amount of savings period by period, the different welfare change from the same policy is the result of
Fig. 4. Welfare changes: Japan.
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Fig. 5. Welfare changes: EEX.
different consumption levels. As indicated in the last section, the reaction to policy shocks through myopic expectation is not justified. Therefore, welfare analysis based on myopic expectation is not accurate. Figs. 3]6 indicate that the differences in welfare changes between the two algorithms warrant serious reconsideration of recursive algorithm for policy analysis purposes. To examine the long-run steady-state properties of the two algorithms, the time paths of the price ratio of capital to labor Ž rrw ., for the four regions in reference run, are depicted in Figs. 7]10. Since the labor supplies are exogenous and
Fig. 6. Welfare changes: China.
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Fig. 7.
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rrw ratio: USA.
identical in both algorithms, the differences in rrw are solely caused by different investment amounts: one is based on myopic expectation; another is based on forward-looking optimization. Several observations are worth noting. First, neither algorithm converges to the base year benchmark calibrated level of 1.0. The results invalidate the strong assumption of extrapolating an initial calibrated equilibrium to a long time horizon. Second, rrw ratios from the coupled algorithm show ‘turnpike’ patterns in three out of four regions. In three regions, rrw ratios are stabilized after initial few
Fig. 8.
rrw ratio: Japan.
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Fig. 9.
rrw ratio: EEX.
periods. On the dual side of the phenomena, the capitalrlabor ratios are stabilized over time. China is an exception. In the EPPA model, China is assumed to have a high growth of labor in efficiency units. For the time horizon of the model, China does not settle on a steady-state path. Third, the paths of rrw ratios of the recursive and coupled algorithms can be quite different. Myopic expectation sends wrong signals for saving decisions in general, especially when the calibrated base year data are far off the long-run steady-state path. However, if the calibrated data
Fig. 10.
rrw ratio: China.
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is close to the steady-state path, as happened in the case of Japan, the results from the myopic expectations and forward looking expectations can be close. By comparing the results from recursive and coupled algorithms implemented in the EPPA model, we clearly demonstrate the improvements of coupled algorithm over recursive algorithm. By showing that the long-run steady-state differs from the base year, we basically refute the strong assumption that base year is on the steady state, which is the basis of many CGE modeling endeavors. Therefore, a coupled algorithm that combines traditional CGE model with optimal growth model is a distinct improvement if the model is intended for a long time horizon.
5. Conclusions The coupled algorithm for solving long-run large-scale CGE models suggested in this paper provides an effective method to achieve the dual tasks of policy analysis and economic forecasting. By combining computable general equilibrium and dynamic programming approaches, the economic modeling can fulfill the tasks that cannot be done by a single approach alone. The demonstration on the EPPA model as shown in this paper proves the point. Although saving and investment decisions are among the most important intertemporal decisions in dynamic economic models, economic agents’ interactions over time are not confined to saving and investment. The coupled algorithm can capture other dynamic mechanisms, at the expense of more model complexity and longer computing time. For example, the balance of current account in multiregion CGE models is one of the model closure rules and is set exogenously region by region. The question of optimal international capital flows, perhaps constrained by intertemporal regional budgets, often need to be addressed by economic models. The DEPPA model described in Appendix A can provide such type of analysis. By dropping ‘island economy’ assumptions, the DEPPA model can be used to solve for optimal international capital flows under the additional assumptions on intertemporal budget constraints and the Negishi weights. Along with the saving and investment decisions, these capital flow variables can be fed into the static CGE model as a set of closure rules. Another potential application of the coupled algorithm is the calculation of forward looking expectations created by price or quantity shocks of energy supply. Because the large-scale models like the EPPA are focused on long-term relationship between economic growth and energy demand, the forward looking mechanism in detailed energy sectors are crucial for better description of long-run energy demand patterns. In the DEPPA model, if the energy input EŽ t . becomes endogenous and is consistent with the EPPA calibrations, the supply of EŽ t . is affected by expected shocks on EŽ t q s . Ž s ) 0.. Then EŽ t . can be put into the decision set fed into the EPPA along with investment. Finally, many ‘learning by doing’ mechanisms can be incorporated into the CGE modeling framework by the coupled algorithm. The coupled algorithm is not a new methodology per se. It calls for combination
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of the strengths of different modeling approaches to set better models. Given fast-improving computing technologies and accumulating modeling experiences, the principle of coupling or combination should have more and more applications. For CGE modeling practice, the coupled algorithm is an effective way to resolve long-run dynamic issues.
Acknowledgements The research is supported by the Joint Program on the Science and Policy of Global Change at MIT. I thank R.S. Eckaus, A.D. Ellerman, H.D. Jacoby and an anonymous referee for their comments and suggestions on the earlier version of this paper.
Appendix A The DEPPA model for coupling the EPPA model: t 0 qT
Max
H I Ž r ,t .
t0
R
Ý f Ž r . U w c Ž r ,t .x eyd t dt
Ž A-1.
rs1 a
b 1yayb s.t. Y Ž r ,t . s AŽ r ,t . L Ž r ,t . K Ž r ,t . E Ž r ,t . C Ž r ,t . s Y Ž r ,t . y I Ž r ,t . y EX Ž r ,t . q IM Ž r ,t .
Ž A-2. Ž A-3.
K˙Ž r ,t . s I Ž r ,t . y d K Ž r . K Ž r ,t .
Ž A-4.
c Ž r ,t . s C Ž r ,t . rPop Ž r ,t .
Ž A-5.
CAŽ r ,t . s EX Ž r ,t . y IM Ž r ,t .
Ž A-6.
˙ Ž r ,t . s CAŽ r ,t . q l Ž t . NFAŽ r ,t . NFA R Y Ž r ,t . lŽ t . s Ý rR Ž . rs1 K r ,t
Ž A-7.
< CAŽ r ,t . < F d1Y Ž r ,t .
Ž A-9.
Ž A-8.
0 - d1 - 1
NFAŽ r ,T . s 0
Ž A-10.
R
Ý CAŽ r ,t . s 0
Ž A-11.
rs1 R
Ý fŽ r . s R
fŽ r . ) 0
Ž A-12.
rs1
AŽ r ,t 0 . s
Ym Ž r ,t 0 . a
b
L Ž r ,t 0 . K Ž r ,t 0 . E Ž r ,t 0 .
Ž A-13.
1yayb
d ) 0, 0 - d K - 1, 0 - a - 1, 0 - b - 1,
aqb-1
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Definitions of variables and parameters: U w cŽ r,t .x: Y Ž r,t .: AŽ r,t .: LŽ r,t .: K Ž r,t .: EŽ r,t .: C Ž r,t .: I Ž r,t .: EX Ž r,t .: IM Ž r,t .: cŽ r,t .: PopŽ r,t .: CAŽ r,t .: NFAŽ r,t .: lŽ t .: fŽ r .:
Utility function of region r at time t; Production function; Total factor productivity trend and adjustment factor; Labor supply in efficient units; Capital stock; Exogenous energy demand, calibrated with EPPA; Aggregate consumption function; Investment function; Total exporting; Total importing; Per capital consumption; Population; Current account balance; Net foreign assets; Average returns to capital; Social welfare weights.
References Australian Bureau of Agriculture and Resource Economics, ABARE. The MEGABARE Model: Interim Documentation. Mimeo, February, 1996. Beavis, B., Dobbs, I.M., 1990. Optimization and Stability Theory for Economic Analysis. Cambridge University Press, New York. Brook, A., Kendrick, D., Meeraus, A., 1992. GAMS: A User’s Guide. The Scientific Press, San Francisco. Dirkse, S.P., Ferris, M.C., 1995. The PATH Solver for GAMSrMCP. Mimeo, October. Drud, A.S., 1992. CONOPT: a large-scale GRG code. ORSA J. Comput. 6, 207]216. Hargerber, A.C., 1962. The incidence of the corporation income tax. J. Polit. Econ. 70, 215]240. Harrison, G., Rutherford, T.F., 1997. Burden Sharing, Joint Implementation, and Carbon Coalition. Mimeo, June. Lau, L.J., 1984. Numerical specification of applied general equilibrium models: estimation, calibration, and data: comments. In: Scarf, H.E., Shoven, J.B. ŽEd.., Applied General Equilibrium Analysis. Cambridge University Press, New York, pp. 127]137. Meade, J.E., 1955. Trade and Welfare: Mathematical Supplement. Oxford University Press, Oxford. Montgomery, D., Bernstein, P., Rutherford, T., 1997. Does Trade Matter? Models Specification and Impacts on Developing Countries. Mimeo, June. OECD, 1992. The Economic Costs of Reducing CO 2 Emissions. Special Issue, OECD Economic Studies, No. 19 Winter. Rutherford, T.F., 1993. Applied General Equilibrium Modeling with MPSGE as GAMS Subsystem: An Overview of the Modeling Framework and Syntax. Computational Economics. Scarf, H.E., Shoven, J.B., 1984. Applied General Equilibrium Analysis. Cambridge University Press, New York. Shoven, J.B., Whalley, J., 1992. Applying General Equilibrium. Cambridge University Press, New York. Takayama, A., 1985. Mathematical Economics. Cambridge University Press, New York. Yang, Z., Eckaus, R.S., Ellerman, A.D., Jacoby, H.D., 1996. The MIT Emissions Prediction and Policy Analysis ŽEPPA. Model. MIT Joint Program on the Science and Policy of Global Change Report No. 6.
A coupling algorithm for computing large-scale dynamic computable general equilibrium models Zili YangU Department of Energy, En¨ ironmental and Mineral Economics, The Pennsyl¨ ania State Uni¨ ersity, 221 Walker Bldg., Uni¨ ersity Park, PA 16802, USA
Abstract Large-scale dynamic computable general equilibrium ŽCGE. models are widely used for policy evaluation and economic forecasting in the studies of global environmental problems, such as climate change. Investigating these issues requires that CGE models have long time spans. Widely used dynamic mechanisms in CGE modeling, such as myopic expectations and calibration on ‘balanced growth’ path, are inadequate for such a long time scale. To improve the performances of long-term CGE models, a coupled algorithm that combines CGE modeling with optimal growth modeling approach, is introduced in this paper. The caveats of existing methods and improvements from the coupled algorithm are explored analytically as well as exemplified in the EPPA model, one large-scale dynamic CGE model with long time horizon. Numerical results from the EPPA model show the superiority of the coupled algorithm over existing treatments of dynamics in CGE models. Q 1999 Elsevier Science B.V. All rights reserved. JEL classifications: C61; C63; C68 Keywords: CGE modeling; Optimal growth; Modeling methods; Economic forecasting
1. Introduction Computable general equilibrium ŽCGE. models are widely used for empirical economic analysis and policy evaluations. Thanks to the growing capacity of U
Tel.: q1-814-863-7071; fax: q1-814-863-7433; e-mail: [email protected]
0264-9993r99r$ - see front matter Q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 2 6 4 - 9 9 9 3 Ž 9 9 . 0 0 0 1 0 - 3
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computing technologies, CGE models have evolved from elegant stylized small models wfor example: Meade’s model on international trade ŽMeade, 1955. and Harberger’s model on tax incidence ŽHargerber, 1962.x for highlighting economic theories to large, multi-sector, multi-region models that simulate economic relationships in great detail. CGE modeling also has become an important field of research. The basic methodology of CGE modeling as well as a plethora of CGE models developed in the 1970s and 1980s are brilliantly summarized by Shoven and Whalley Ž1992. as well as Scarf and Shoven Ž1984.. Most CGE models are static models built on a given data set. Economic analyses are based on comparisons between a baseline scenario and ‘counter factual’ scenarios induced by endogenous or exogenous shocks. Due to the richness of sectoral andror regional breakdown, CGE models are good analytical tools for trade policy and tax policy analysis. Such issues frequently involve the policy impacts on the welfare of different sectors, population groups and regions. The CGE modeling approach receives its fair share of criticism, especially from econometricians wfor example, see Lau Ž1984.x. The major criticism is focused on the parametric specifications of CGE models, which lack adequate statistical justification. Thus the results from CGE models are potentially compromised by their sensitivity to parameter assumptions. Despite those criticisms, CGE models remain effective analytical tools for indicative purposes. Very few large-scale CGE models are dynamic because of their intrinsic complexity, which will be outlined in the next section. Nonetheless, attempts have been made to build such models. Increasing concern about long-term impacts of some global environmental problems, such as greenhouse gas ŽGHG. emissions, have prompted the building of large-scale dynamic CGE models. Those models usually serve at least two purposes: they forecast long-run economic growth and the evolution of energy use patterns; they also support analysis of costs of carbon control policies, based on their richness of sectoral and regional breakdown. An early example of such models is OECD’s GREEN model ŽOECD, 1992.. More recently, several large-scale dynamic CGE models have been built for studying global environmental issues. They include MIT’s EPPA model ŽYang et al., 1996. which uses the GREEN data set, and the Australian Bureau of Agriculture and Resource Economy’s MEGABARE model ŽABARE, 1996., and the CETM model ŽMontgomery et al., 1997.. This list is not complete and more models are coming. The dynamic mechanisms of these models are different. In principle, they are built on either on myopic expectation or calibrated on some ad hoc paths. In those models, the most important intertemporal economic relationship } namely, saving, investment and capital formation } lacks any forward-looking mechanism. Instead, strong, exogenous assumptions are imposed on the intertemporal growth path. As a result, the forecasting capability of these dynamic CGE models is limited. In this paper, I propose a new algorithm to overcome the above shortcomings in long-run dynamic CGE models. The fundamental structure of the algorithm involves the coupling of a static CGE model with a matching Ramsey-type optimal growth model to replace the recursive dynamic mechanism in CGE models.
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Intertemporal optimal decisions on saving and investment are made in the optimal growth model, which feed into the static CGE model. This coupled CGE model has the essential desired properties of both a forward looking optimal growth model and a static CGE model. Namely, a time-consistent growth path to forecast future economy and a rich sectoral breakdown to conduct relevant policy analysis. Computationally, the coupled model is efficient. It has the same order of computing time with corresponding recursive dynamic model, if the initial searching condition is properly set. The remainder of this paper is organized as follows. In the next section, a brief summary and critique of recursive dynamic approach in CGE modeling and the rational for the coupling algorithm are presented; in Section 3, the new algorithm for coupling a static CGE model with a forward-looking optimal growth model is introduced; in Section 4, the algorithm is implemented into the EPPA model ŽYang et al. Ž1996.., a large-scale recursive dynamic model. A detailed comparison between old and new algorithms is also presented here. In the last section, the direction of further work is indicated.
2. Caveats in recursive-dynamic CGE models Most empirical CGE models are static models. One reason for this is the data requirements of dynamic CGE models. These are calibrated models, based on disaggregate input]output tables. It is virtually impossible to obtain input]output tables of given regions for consecutive periods as modelers desire. Another reason is that the static CGE models can fulfill many policy analysis tasks, using counterfactual simulations on a calibrated equilibrium to probe the impact of different price and quantity shocks. However, if a CGE model is designed not only for policy analysis but also for economic forecasting, where no data for calibration are available, the time dimension and certain mechanisms of intertemporal dynamics have to be included in the model. In practice, recursive mechanism is widely adopted by CGE modelers. A CGE model based on recursive dynamic principles is actually a series of static models solved sequentially. Because the calibration is only imposed on the initial period, the variables in subsequent periods in the reference path are calculated with changing exogenous and endogenous endowments. The outcomes from subsequent periods with no policy intervention can be treated as quantity Žendowment. shocks to the calibrated initial period. In other words, the reference scenario in a recursive dynamic CGE model is a series of ‘counter-factual’ equilibria based on the initial period. While endowment changes in subsequent periods can be defined as exogenous or endogenous, very strong assumptions about exogenous endowment changes, such as capital formation, may be required. Many important economic activities are intrinsically forward-looking. Current saving decisions of people are partially based on their expectations of future returns; future resource availability and expected policy shocks affect people’s current consumption patterns. The recursive approach in CGE models is not
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designed to deal with these issues. When such issues are encountered in recursivedynamic CGE models, special compromising assumptions are made to resolve the problems. In this paper, we direct our attention to saving and investment problem only and more complicated dynamic issues are indicated as future research direction, following the methodology provided in this paper. In CGE modeling practice, intertemporal decisions on saving and investment are based on conventional ad hoc assumptions. Two commonly adopted assumptions are either a ‘balanced growth path’ or myopic expectations. Under the ‘balanced growth path’ assumption, the ratios of the capital stock to the labor supply in future periods are the same as the ratio in the calibrated base period. If the growth of these two primary factors are disproportional in physical units, efficiency factors are applied to one or both primary inputs to maintain a constant ratio over time between the capital stock and the labor supply in efficiency units. The GREEN model ŽOECD, 1992. adopted this approach. Recently, there are some fully forward-looking CGE models that have been built on such a ‘balanced growth’ assumption ŽHarrison and Rutherford, 1997.. In these models, the economy is calibrated on a balanced growth path for the whole time horizon. Thus, the model generates an Arrow]Debreu economy in which commodities have both time and sector dimensions. The reactions of the economic system to the counter factual shocks are forward-looking. Alternatively, saving and investment decisions in a recursive CGE model can be based on certain myopic expectations. In each period, the saving and investment decision is determined by evaluation of lagged dual variables. Namely, Sa¨ Ž t . s In¨ Ž t . s f w r Ž t y 1 . rw Ž t y 1 . ,Y Ž t .x
Ž1.
Here, r is rate of returns to capital, w is wage rate and Y is the income of the representative consumer. This assumption makes the economy adjust its investment levels in each period according to the relative scarcity of capital in previous period. The EPPA model ŽYang et al., 1996. uses this assumption. Although the above two assumptions are less than desirable for describing saving and investment behavior in CGE models, they are still acceptable in many circumstances. Particularly, when the time horizon is short and the economy to be modeled is stable within the time horizon, the assumptions may be good approximations for the simulation of intertemporal dynamics. When the time horizon is short, say 5]10 years, the ratio of capital stock to labor does not change significantly, as long as there are no implausible and drastic changes in demographic composition, large-scale retirement of capital or sudden and very large increases in investment. For those economies that are in transition, or not stable, the balanced growth assumption is not proper even for a short time horizon, however. For example, economic reform in China resulted in a fast-growing economy and caused fundamental structural changes in the economy. The collapse of the Soviet political system also caused partial collapse of Russian economy. In both situations, the economy to be modeled is not on a balanced growth path. If the calibrated base
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period is followed by such events, the balanced growth path assumption can lead to large errors. In general, both recursive dynamic and ‘balanced growth’ path assumptions in CGE modeling are not consistent with the theories of economic growth. In effect, the notion of balanced growth is equivalent to the term ‘equilibrium’ in a static setting. One major premise of CGE modeling is that the calibrated base period represents an equilibrium state.1 In a recursive dynamic CGE model, all subsequent periods would be proportional ‘copies’ of the initial equilibrium except for exogenously imposed changes in policies andror resource availabilities. However, from a long-run perspective, the base period is no more than an arbitrary observation on growth path. It is implausible to assume that an economy, during an arbitrary initial period, is located on its long-run balanced growth path. The logic behind the assumption is contradictory to the economic theories explaining optimal economic growth patterns, such as turnpike theorems wsee Takayama Ž1985, ch. 7. for a summary of turnpike theoremsx. According to these theories, starting from an arbitrary base period, an optimal saving-investment plan can drive the economy to the long-run balanced growth path quickly. In general, the optimal capital labor ratio along the balanced growth path is different from that of the initial period. Therefore, the fundamental assumption on the calibrated equilibrium in the base period is not justified for the long-run growth path. The assumption of myopic expectations also has some severe problems. First, unless the economy is on the balanced growth path, myopic expectation is not consistent with fully foresighted expectations. Since the steady-state assumption is not true for any arbitrary initial period, the myopic expectation is not necessarily intertemporally optimal. Second, in counter-factual simulations, policy shocks always cause changes in relative prices. As a result, the price signals for the myopic expectations also change. However, these price change in current period are not consistent predictors for long-run capital formation. For example, if a policy shock in one period causes the relative price of capital to increase, that does not necessarily imply that long-run demand for capital will increase. Or, in the next period, a similar policy shock might cause the relative price of capital to decrease, yet it does not necessarily mean that capital formation should slow down. However, they would do so, according to Eq. Ž1.. In summary, the impact of current policy shocks on the long-run capital formation through the prism of myopic expectation is inconclusive, even arbitrary. Furthermore, the savings behavior of myopic expectations changes the disposable income of the representative agent. If a policy shock changes the size of the consumption ‘pie’ arbitrarily, the welfare analysis of the policy impact is questionable. Finally, the pure time preference of the representative consumer, which need not be specified in a model based on myopic expectations, does not affect his saving and investment decision. Therefore, saving and investment decisions are not consistent with optimal saving and investment decisions under a given pure time 1
In modeling practice, different types of ‘dis-equilibria’ can be built into the base period ad hoc.
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preference. As a result, the intertemporal welfare analysis of policy impacts lacks a clean foundation. These considerations lead to the following conclusions. While the assumptions of balanced growth path and myopic expectations are reasonable and acceptable for short-run dynamic CGE models, they are not satisfactory for long-run CGE models. If the long-run CGE model is to be used for both policy analysis and economic forecasting, a better treatment of long-run dynamics is required. The coupling algorithm, suggested in this paper, overcomes shortcomings of recursive dynamic or ‘balanced growth’ path assumptions in CGE modeling.
3. A coupling algorithm for long-run CGE models To achieve the dual tasks of policy analysis and economic forecasting, a long-run large-scale CGE model has to depart from the traditional assumptions that the initial period is on a balanced growth path. The assumption of equilibrium in the initial period should be used only for benchmark calibration purposes. The mechanisms of intertemporal saving and investment should be guided by the principles of optimal economic growth theories. According to these theories, if in the initial period the economy is not on its balanced growth path, it shifts to this path by adjusting saving and investment and thereby changing its laborrcapital ratio. To reflect this pattern, an efficient approach is to have an optimal growth model, with identical initial conditions and exogenous trend assumptions, interact with the CGE model, and provide the necessary saving and investment decisions for the CGE model. The coupling algorithm suggested in this paper implements this approach. To outline the coupling algorithm, we first express a recursive dynamic CGE model as a mapping of its primary factor inputs Žlabor and capital. to a state of economic equilibrium and Žusing M to indicate myopic. denote it as M w LŽ t ., K M Ž t .x. We assume that LŽ t . is exogenously specified and K M Ž t . is determined from certain recursive rules.2 Within the image of M Ž t ., are prices and quantities of commodities and factor demands. Some important aggregate variables, such as GNP and consumption, are extracted for interaction with the dynamic model. The corresponding optimal growth model, denoted as Dw LŽ t ., K D Ž t ., ID Ž t .x, is a dynamic system with the same exogenous input LŽ t ., the control variable ID Ž t . Žinvestment. and state variables wincluding K D Ž t .x. All state variables in DŽ t . have their counterpart in M Ž t .. The control variable ID Ž t . is interactive with M. M updates its investment based on ID Ž t .. In addition, DŽ t . has a free variable AŽ t . that can be adjusted independent of M. The analytical specification of AŽ t . can be interpreted as a Hicksian neutral technology progress factor. However, its function here is operational, rather than interpretational. As in other simple optimal growth 2 While labor supply, LŽ t ., is exogenous in most large-scale CGE models for various reasons, some models, for example MEGABARE ŽABARE, 1996., do have endogenous labor supply mechanism.
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models, AŽ t . is exogenous. We allow the adjustment of AŽ t . to ensure consistency of state variables between DŽ t . and M Ž t .. The procedure of the coupling algorithm is the following. First, the base period of D is calibrated to the initial condition of M such that YM w LŽ t 0 ., K M Ž t 0 .x s YD w LŽ t 0 ., K D Ž t 0 .x. Here Y is any common state variable in both M and D, GNP for example. Then D is solved for optimal saving and investment based on an initial AŽ t . path. The magnitude IDU Ž t 0 q 1. on the optimal solution path IDU Ž t . is selected to feed into the next period static CGE model M Ž t 0 q 1. as the aggregate saving-investment decision in the static model. Once IM Ž t 0 q 1. s IDU Ž t 0 q 1. is set, we solve the static CGE model for the period t 0 q 1, namely M Ž t 0 q 1.. The endogenous variables, such as GNP, from M Ž t 0 q 1. are then compared with the same variables from the optimal solution path of DU Ž t 0 . in its second period. If the difference between the two is smaller than certain pre-determined tolerance level, we can claim that the forward-looking saving-investment decision in DU Ž t 0 . is consistent with M Ž t 0 q 1.. Otherwise, the whole procedure is repeated from the first step by redefining AŽ t . according to certain tatonnement rules. When the convergence criterion regarding the endogenous variables is met, the algorithm moves to the next period to start all over again. A new optimal growth model starting from t 0 q 1 is defined with YM w LŽ t 0 q 1., K M Ž t 0 q 1.x s YD w LŽ t 0 q 1., K D Ž t 0 q 1.x as the initial condition and a new path of AŽ t . as the initial searching point. Fig. 1 is the schematic flow chart of the above-mentioned procedure. The procedure, with inference to the EPPA model, is illustrated in the flow chart in Appendix A. We claim that this coupling algorithm provides the correct aggregate saving and investment decisions for the CGE model to be coupled. First, M Ž t 0 . and DŽ t 0 . are started from the identical initial conditions for any t 0 in planning horizon of M. Therefore, the optimal intertemporal decisions in D are made upon the identical initial conditions and assumptions about exogenous endowment changes of M. Second, the forward-looking saving-investment decision IDU Ž t 0 q 1. in DŽ t 0 . generates the identical GNPs in period t 0 q 1 in DŽ t 0 . and in M Ž t 0 q 1.. In other words, the decision made in M Ž t 0 q 1. is consistent with the optimal forward-looking expectations from the previous period. On the balanced growth path or not, this decision is time-consistent and optimal regarding to the objective function in DŽ t .. Since we accept DŽ t ., a sequence of forward-looking optimal growth models starting at each period in M Ž t ., as the intertemporal decision maker for M Ž t ., a sequence of static CGE models, we in fact equip M Ž t . with a forward-looking mechanism. The combined outcome from the coupled algorithm has the strength of both CGE models and optimal growth models. The modeled economy is on an optimal growth path all the time for its time consistency. If the time horizon is sufficiently long, the ‘turnpike’ pattern is shown. On the other hand, sectoral demands for primary factors, impacts of policy shocks are still revealed by the static CGE model with rich sectoral and regional details. Also, the welfare analysis of policy impacts over time in the CGE model is consistent when using the pure time preference defined in the optimal growth model. Finally, we should point out that the coupling
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Fig. 1. Programming flow chart of coupled EPPA model Ž D is the dynamic model and M is the myopic submodel..
algorithm enhances forecasting capability of the CGE model to be coupled. Policy analysis tasks are still fulfilled by the CGE model.
4. Implementing the coupling algorithm in the EPPA model To highlight the necessity and efficacy of the coupling algorithm, we employ it in the EPPA model. The EPPA model is a global CGE model with 12 regions linked by bi-lateral trade matrix. In each region, there are eight production sectors and
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four consumer sectors. The model is driven by regionally-specified labor and capital supplies as well as sectoral-specified fixed factor supplies. The planning horizon is from 1985 to 2100 in 5-year intervals. The intertemporal dynamics of saving and investment decisions in the EPPA model now follows some myopic expectation rules similar to Eq. Ž1.. A detailed descriptions of the model is provided by Yang et al. Ž1996.. The EPPA model is an ideal candidate for testing the coupling algorithm because of its dual tasks of economic forecasting and policy analysis for a long time horizon. For executing the coupling algorithm, a matching aggregate Ramsey type optimal growth model is built. For the convenience, this optimal growth model is labeled DEPPA model. The DEPPA model is a optimal growth model with the same 12 regions as the EPPA model. Each region uses labor, capital and energy as inputs to produce an aggregate good called GDP. The objective function of the optimization problem is a social welfare function consisting of a weighted sum of intertemporal utility functions of 12 regions. Trade flows are allowed in the DEPPA model with the trade balances equivalent to interregional capital flows because only an aggregate homogenous good is produced here. The formal description of the DEPPA model is in Appendix A. Two underlining assumptions in the EPPA model make the setting up of the DEPPA model relatively easy. First, in the multi-layer CES production structure of the EPPA model, the current value of the substitution elasticity between labor and the bundle of capital, energy and fixed factor is 1.0. Thus the aggregate Cobb]Douglas production function in the DEPPA model is a good structural approximation of the EPPA model. Second, one of the closure rules in the EPPA model sets the balance on current account equal to zero Ži.e. each region is required to maintain trade balance .. The equivalent assumption in the DEPPA model is that capital flows between regions are zero. Therefore, the DEPPA model behaves like 12 ‘island’ economies without interactions among each other. The social welfare weights can be set arbitrarily in the DEPPA model. The assumption reduces the complexity of computation.3 The coupling algorithm is programmed in the GAMS language ŽBrook et al., 1992.. The EPPA model is solved by PATH solver ŽDirkse and Ferris, 1995. under MPSGE platform ŽRutherford, 1993. and the DEPPA model is solved by CONOPT solver ŽDrud, 1992.. The coupling algorithm is implemented by switching solvers and sharing information between intermediate iterations. The coupled model takes 3 h to solve on a Pentium Pro 200 computer with a fixed number Ž12. of tatonnement iterations between DEEPA and EPPA in each period. The original EPPA takes 10 min on the same machine. 3
In a detailed CGE model, the trade flows between regions can be enormous under the assumption of trade balance. We assume that the trade pattern does not have direct impact on domestic saving and investment behavior. Nevertheless, the more general assumption on capital flows, as defined in Appendix A, will be useful for many research and modeling purposes. We will indicate them later in the paper.
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For comparison purposes, four sets of simulations on the coupled model are conducted here. They are: Ži. Žii. Žiii. Živ.
a baseline scenario based on myopic expectations in saving decisions; a baseline scenario based on the coupled algorithm; a policy shock scenario based on myopic expectations in saving decisions; and the same policy shock scenario based on the coupled algorithm.
The policy adopted in Žiii. and Živ. is a version of so-called ‘AOSIS’ policy on carbon emission control. It requires the OECD region to curb its carbon emissions levels from 2010 onward at 80% of its 1990 level. This policy scenario is a good device for testing the properties of a global model linked by trade matrix, because of its draconian feature. To simplify the presentation and analysis, data and graphs are extracted from four regions out of total 12 regions in the EPPA model. To examine the convergence properties of the coupled algorithm, the maximum relative differences in GNP between the EPPA and DEPPA models are reported in Table 1. The definition of table entries is the following:
Max r
½
GNPM Ž r . y GNPD Ž r . GNPM Ž r .
5
Ž2.
where r is the index for regions. From Table 1 we can conclude that the two models are very close to each other after a few iterations. 4 Because the EPPA model is a complex non-linear system, convergence using the tatonnement method is not monotonic. The entries in Table 1 show the pattern. The rate of convergence can be improved by refinement of tatonnement method. Nevertheless, given the fact that the maximal differences in GNP in two models are smaller than 0.5% in most periods, the two models are consistent in the periods when making interactive saving and investment decisions. In addition, since investment is only a fraction of total GNP, the numerical errors reflected in Table 1 on saving and investment decisions are negligible. Fig. 2 plots the welfare changes of the coupled algorithm relative to that of recursive algorithm for USA, Japan ŽJPN., energy exporting countries ŽEEX. and China ŽCHN., respectively. In the main, welfare improvements are observed in most periods for all regions. Since the welfare indices are functions of consumption Žwhich is personal income net of saving., the welfare index is an indirect measure of savings. In addition, DEPPA’s maximand is a function of the intertemporal utility of regions, which in turn, are functions of consumption. Saving decisions affect welfare levels now and in the future. The saving decisions in recursive-dynamic CGE models are not based upon intertemporal welfare maximization. Therefore, it is unlikely that the recursive CGE model is on the optimal growth path. As the results, welfare indices from the recursive CGE model are inferior to 4
To test the convergence properties, a fixed number Ž12. of iterations are used instead of stopping by using a specified convergence criterion.
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Table 1 Maximum relative deviations of GNP between EPPA and DEPPA models Ž%. Iteration period
1
2
3
4
5
6
7
8
9
10
11
12
1990 1995 2000 2005 2010 2015 2020 2025 2030 2035 2040 2045 2050 2055 2060 2065 2070 2075 2080 2085 2090 2095 2100
14.26 13.25 12.54 10.25 7.44 5.19 4.01 4.22 4.52 4.82 5.09 5.29 5.42 6.01 6.68 7.13 7.28 7.16 6.82 6.32 5.79 5.26 5.05
13.65 10.34 8.49 6.81 4.57 2.98 1.96 2.15 2.30 2.43 2.52 2.84 3.38 3.90 4.34 4.66 4.82 4.80 4.64 4.38 4.06 3.77 3.51
11.87 7.31 4.52 3.34 1.70 1.06 1.36 1.73 2.10 2.38 2.63 2.88 3.03 3.09 3.06 2.93 2.76 2.52 2.49 2.45 2.37 2.32 2.22
10.23 4.91 1.74 0.91 0.98 2.03 2.23 2.58 2.96 3.23 3.40 3.66 3.86 3.95 3.98 3.90 3.77 3.58 3.35 3.07 2.80 2.50 2.25
8.79 3.19 0.66 0.92 0.80 1.38 1.33 1.38 1.44 1.46 1.46 1.45 1.45 1.53 1.61 1.66 1.70 1.74 1.77 1.74 1.71 1.66 1.58
7.54 2.03 0.51 1.01 0.73 0.64 0.43 0.51 0.80 1.11 1.36 1.52 1.60 1.58 1.50 1.32 1.11 0.86 0.66 0.62 0.58 0.53 0.46
6.45 1.27 0.42 0.69 0.43 0.73 1.04 1.40 1.77 2.06 2.40 2.63 2.75 2.79 2.76 2.58 2.39 2.13 1.86 1.52 1.29 0.98 0.77
5.52 1.07 0.30 0.28 0.33 0.81 0.94 1.10 1.25 1.34 1.38 1.42 1.50 1.54 1.59 1.58 1.56 1.50 1.44 1.29 1.16 1.03 0.88
4.71 0.90 0.23 0.22 0.22 0.36 0.24 0.17 0.23 0.39 0.56 0.67 0.69 0.66 0.56 0.39 0.25 0.21 0.22 0.27 0.33 0.40 0.43
4.02 0.76 0.18 0.12 0.09 0.16 0.38 0.67 0.99 1.33 1.61 1.79 1.88 1.87 1.80 1.61 1.41 1.14 0.91 0.59 0.43 0.25 0.15
3.42 0.65 0.14 0.11 0.09 0.37 0.54 0.74 0.92 1.05 1.18 1.26 1.32 1.35 1.37 1.30 1.22 1.13 0.98 0.78 0.67 0.50 0.40
2.91 0.55 0.10 0.07 0.11 0.28 0.29 0.28 0.22 0.15 0.13 0.17 0.16 0.15 0.18 0.23 0.26 0.28 0.35 0.40 0.34 0.37 0.31
those from the coupled model. In summary, welfare indices, ceteris paribus, are import criteria for evaluating the coupling algorithm. We should note that the changes in the second period for USA and CHN are large. The reason is that the initial saving rates is very high in China Ž30%. and is very low in USA Ž13%. in the EPPA model. In the second period the optimal saving rate from the DEPPA model is much lower for China and much higher for USA than in the base period. Such saving rates from the DEPPA alter consumption and cause drastic differences between the recursive EPPA results and forward-looking DEPPA results. The relatively low saving rate in the second period decreases the welfare levels in subsequent few periods in China. The relatively higher saving rate in the second period increases the welfare levels in almost all future periods in USA. Although a single period saving decision does not necessarily bear direct causality for all future welfare levels, we can still conclude that the saving rates from a given period need not be optimal for the long-run economic growth. Table 2 reports the present values of welfare indices discounted at 3% annually Žthe rate of pure time preference used in the DEPPA. from recursive and coupled algorithms, respectively. Method A is calculated by setting the base period at 1 and summing up all 24 periods. Method B provides the results from Method A net of the values of the
466
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Fig. 2. Welfare changes in reference scenario Žcoupled vs. recursive..
second period. The values of welfare indices from the coupled algorithm are consistently higher in Method B and are higher for three regions except USA in Method A. The lower present value of welfare index for USA in Method A is because of higher saving in the second period Žas shown in Fig. 2.. Nonetheless, the overall improvements on welfare from coupled algorithm over recursive algorithm are there, even for USA. The two methods of calculating present value of welfare indices show that coupled algorithm is better than recursive algorithm by so-called ‘catching up’ criterion Žsee Beavis and Dobbs, 1990.. Figs. 3]6 show welfare losses Žor gains. due to the AOSIS policy in four regions. USA and Japan face carbon emission constraints directly. As the results, demands for fossil fuels decline and welfare levels decrease. Energy exporting countries lose a major buyer ŽOECD. of their energy exporting commodities. They experience an even larger welfare loss because of the change in the terms of trade. In contrast, China has a very moderate welfare gain due to the change in the terms of trade under the AOSIS scenario. The results show that the recursive and coupled algorithms have different implications on the costs for carbon control policies. In the cases of the USA and
Table 2 The present value of welfare indices USA
JPN
EEX
CHN
Method A
Recursive Coupled
2.09700 2.05194
2.17133 2.18470
2.77094 2.80478
3.35900 3.42986
Method B
Recursive Coupled
1.58294 1.58669
1.65314 1.67170
2.02454 2.05619
2.49653 2.50270
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Fig. 3. Welfare changes: USA.
Energy Exporting Countries, the AOSIS policy incurs lower costs from the recursive algorithm than they are from the coupled algorithm. On the other hand, the AOSIS policy incurs higher costs from the recursive algorithm than they are from the coupled algorithm in Japan. In case of China, the gains from the AOSIS policy is lower from the recursive algorithm. The magnitude of the differences in the policy generated welfare changes between the two algorithms are significant. Since the only structural difference between the two algorithms is the amount of savings period by period, the different welfare change from the same policy is the result of
Fig. 4. Welfare changes: Japan.
468
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Fig. 5. Welfare changes: EEX.
different consumption levels. As indicated in the last section, the reaction to policy shocks through myopic expectation is not justified. Therefore, welfare analysis based on myopic expectation is not accurate. Figs. 3]6 indicate that the differences in welfare changes between the two algorithms warrant serious reconsideration of recursive algorithm for policy analysis purposes. To examine the long-run steady-state properties of the two algorithms, the time paths of the price ratio of capital to labor Ž rrw ., for the four regions in reference run, are depicted in Figs. 7]10. Since the labor supplies are exogenous and
Fig. 6. Welfare changes: China.
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Fig. 7.
469
rrw ratio: USA.
identical in both algorithms, the differences in rrw are solely caused by different investment amounts: one is based on myopic expectation; another is based on forward-looking optimization. Several observations are worth noting. First, neither algorithm converges to the base year benchmark calibrated level of 1.0. The results invalidate the strong assumption of extrapolating an initial calibrated equilibrium to a long time horizon. Second, rrw ratios from the coupled algorithm show ‘turnpike’ patterns in three out of four regions. In three regions, rrw ratios are stabilized after initial few
Fig. 8.
rrw ratio: Japan.
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Fig. 9.
rrw ratio: EEX.
periods. On the dual side of the phenomena, the capitalrlabor ratios are stabilized over time. China is an exception. In the EPPA model, China is assumed to have a high growth of labor in efficiency units. For the time horizon of the model, China does not settle on a steady-state path. Third, the paths of rrw ratios of the recursive and coupled algorithms can be quite different. Myopic expectation sends wrong signals for saving decisions in general, especially when the calibrated base year data are far off the long-run steady-state path. However, if the calibrated data
Fig. 10.
rrw ratio: China.
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is close to the steady-state path, as happened in the case of Japan, the results from the myopic expectations and forward looking expectations can be close. By comparing the results from recursive and coupled algorithms implemented in the EPPA model, we clearly demonstrate the improvements of coupled algorithm over recursive algorithm. By showing that the long-run steady-state differs from the base year, we basically refute the strong assumption that base year is on the steady state, which is the basis of many CGE modeling endeavors. Therefore, a coupled algorithm that combines traditional CGE model with optimal growth model is a distinct improvement if the model is intended for a long time horizon.
5. Conclusions The coupled algorithm for solving long-run large-scale CGE models suggested in this paper provides an effective method to achieve the dual tasks of policy analysis and economic forecasting. By combining computable general equilibrium and dynamic programming approaches, the economic modeling can fulfill the tasks that cannot be done by a single approach alone. The demonstration on the EPPA model as shown in this paper proves the point. Although saving and investment decisions are among the most important intertemporal decisions in dynamic economic models, economic agents’ interactions over time are not confined to saving and investment. The coupled algorithm can capture other dynamic mechanisms, at the expense of more model complexity and longer computing time. For example, the balance of current account in multiregion CGE models is one of the model closure rules and is set exogenously region by region. The question of optimal international capital flows, perhaps constrained by intertemporal regional budgets, often need to be addressed by economic models. The DEPPA model described in Appendix A can provide such type of analysis. By dropping ‘island economy’ assumptions, the DEPPA model can be used to solve for optimal international capital flows under the additional assumptions on intertemporal budget constraints and the Negishi weights. Along with the saving and investment decisions, these capital flow variables can be fed into the static CGE model as a set of closure rules. Another potential application of the coupled algorithm is the calculation of forward looking expectations created by price or quantity shocks of energy supply. Because the large-scale models like the EPPA are focused on long-term relationship between economic growth and energy demand, the forward looking mechanism in detailed energy sectors are crucial for better description of long-run energy demand patterns. In the DEPPA model, if the energy input EŽ t . becomes endogenous and is consistent with the EPPA calibrations, the supply of EŽ t . is affected by expected shocks on EŽ t q s . Ž s ) 0.. Then EŽ t . can be put into the decision set fed into the EPPA along with investment. Finally, many ‘learning by doing’ mechanisms can be incorporated into the CGE modeling framework by the coupled algorithm. The coupled algorithm is not a new methodology per se. It calls for combination
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of the strengths of different modeling approaches to set better models. Given fast-improving computing technologies and accumulating modeling experiences, the principle of coupling or combination should have more and more applications. For CGE modeling practice, the coupled algorithm is an effective way to resolve long-run dynamic issues.
Acknowledgements The research is supported by the Joint Program on the Science and Policy of Global Change at MIT. I thank R.S. Eckaus, A.D. Ellerman, H.D. Jacoby and an anonymous referee for their comments and suggestions on the earlier version of this paper.
Appendix A The DEPPA model for coupling the EPPA model: t 0 qT
Max
H I Ž r ,t .
t0
R
Ý f Ž r . U w c Ž r ,t .x eyd t dt
Ž A-1.
rs1 a
b 1yayb s.t. Y Ž r ,t . s AŽ r ,t . L Ž r ,t . K Ž r ,t . E Ž r ,t . C Ž r ,t . s Y Ž r ,t . y I Ž r ,t . y EX Ž r ,t . q IM Ž r ,t .
Ž A-2. Ž A-3.
K˙Ž r ,t . s I Ž r ,t . y d K Ž r . K Ž r ,t .
Ž A-4.
c Ž r ,t . s C Ž r ,t . rPop Ž r ,t .
Ž A-5.
CAŽ r ,t . s EX Ž r ,t . y IM Ž r ,t .
Ž A-6.
˙ Ž r ,t . s CAŽ r ,t . q l Ž t . NFAŽ r ,t . NFA R Y Ž r ,t . lŽ t . s Ý rR Ž . rs1 K r ,t
Ž A-7.
< CAŽ r ,t . < F d1Y Ž r ,t .
Ž A-9.
Ž A-8.
0 - d1 - 1
NFAŽ r ,T . s 0
Ž A-10.
R
Ý CAŽ r ,t . s 0
Ž A-11.
rs1 R
Ý fŽ r . s R
fŽ r . ) 0
Ž A-12.
rs1
AŽ r ,t 0 . s
Ym Ž r ,t 0 . a
b
L Ž r ,t 0 . K Ž r ,t 0 . E Ž r ,t 0 .
Ž A-13.
1yayb
d ) 0, 0 - d K - 1, 0 - a - 1, 0 - b - 1,
aqb-1
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Definitions of variables and parameters: U w cŽ r,t .x: Y Ž r,t .: AŽ r,t .: LŽ r,t .: K Ž r,t .: EŽ r,t .: C Ž r,t .: I Ž r,t .: EX Ž r,t .: IM Ž r,t .: cŽ r,t .: PopŽ r,t .: CAŽ r,t .: NFAŽ r,t .: lŽ t .: fŽ r .:
Utility function of region r at time t; Production function; Total factor productivity trend and adjustment factor; Labor supply in efficient units; Capital stock; Exogenous energy demand, calibrated with EPPA; Aggregate consumption function; Investment function; Total exporting; Total importing; Per capital consumption; Population; Current account balance; Net foreign assets; Average returns to capital; Social welfare weights.
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