Parameter Uncertainty in CGE Modeling of the Macroeconomic Impact

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622 3.3 Unconditional sensitivity analysis results One of the most important contributions of the uncertainty analysis is the identification of the driving uncertainties, the uncertain parameters that cause the most variance in the outputs of interest. Figure 5 shows the effects of perturbations of each elasticity parameter in its space with random values of other elasticity Tsinghua Science and Technology, October 2006, 11(5): 617-624 parameters selected from their probability density functions. The carbon tax rate and GDP reduction rate calculated by the model when assuming a counterfactual CO2 emission reduction goal of 20% in 2010 were selected as typical TEDCGE outputs in Figs. 5a and 5b. Only the influential parameters are named in Fig. 5. Fig. 5 Effects of perturbations of each elasticity parameter in its space with other elasticities randomized from their probability density functions. Only the most important factors are named. See the notes with Table 2 for explanation of the sector indicators. For a model with 50 uncertain input parameters such as TEDCGE, only a few factors are likely to have a sizeable influence. Figure 5 demonstrates that the TED- CGE model results are sensitive to only a few of the parameters when the critical parameters of various endogenous variables are varied. For example, the uncertainty in the elasticity of substitution between capital and energy in three sectors (Eke(2), Eke(6), and Eke(10)) and the elasticity of substitution between different energy sources in two sectors (Eee(2) and Eee(10)) are the primarily parameters influencing the variances in the carbon tax rate corresponding to a specific carbon emission reduction rate, while the key parameters for the GDP reduction rate are only the elasticity of substitution between the different energy sources in the heavy industry and natural gas sectors (Eee(2) and Eee(10)). A simple measure of importance of an input factor, α, was defined based on the so-called variance of the conditional expectation (VCE) of outcome. Assume that we are interested in describing the importance of an elasiticity parameter α in TEDCGE. The importance of α with regard to the output uncertainty can be assessed by considering the conditional probability distributions of the TEDCGE outputs conditioned on α. The importance of a parameter is assumed to be related to how much it affects the model output. Intuitively, parameter α is important if fixing its value substantially reduces the (conditional) output variance relative to the marginal output variance. Hence, the variance ratio can be used as an appropriate measure of importance. In this study, the magnitude of the VCE relative to the total variance in the output is defined as importance index: IM αi ( Y ) [ ] ( ) [ ] Var ⎡ α E α ⎤ VCE α = ⎣ ⎦ = (5) Var Y Var Y where Varα [E(Y|α)] is the variance of the conditional expectation of the model output of Y, conditioned on α, that is, the variability in E(Y|α) as α varies. In fact, this index is termed the correlation ratio and had been studied and applied by many investigators for sensitivity analysis [19] . The unconditional systematic sensitivity analysis results can be used to calculate the importance index for each uncertain elasticity parameter. Table 2 lists the top ten elasticities with the largest and smallest sensitiveness to the endogenous variables of the carbon tax rate and GDP reduction rate.
WANG Can (王灿) et al:Parameter Uncertainty in CGE Modeling of… 623 Table 2 Order of the sensitivities of elasticities to the CGE model outputs Top ten sensitive elasticities Top ten insensitive elasticities To carbon tax rate To GDP loss rate To carbon tax rate To GDP loss rate Elasticity IMαi Elasticity IMαi Elasticity IMαi Elasticity IMαi E ke(2) 0.36 E ee(2) 0.36 E id(5) 0.02 E ed(9) 0.02 E ee(2) 0.31 E ee(10) 0.19 E ee(5) 0.02 E ee(1) 0.02 E ke(10) 0.26 E ee(7) 0.10 E ke(4) 0.02 E ke(9) 0.02 E ke(6) 0.16 E ke(2) 0.09 E id(9) 0.02 E ed(1) 0.02 E ee(10) 0.12 E ke(10) 0.09 E ed(10) 0.02 E ed(7) 0.03 E ke(3) 0.11 E ee(6) 0.09 E ed(7) 0.03 E ed(4) 0.03 E cl(2) 0.10 E cl(7) 0.09 E ed(9) 0.03 E id(10) 0.03 E ee(6) 0.09 E ee(9) 0.09 E ee(1) 0.03 E cl(1) 0.04 E cl(10) 0.08 E id(4) 0.09 E ke(9) 0.03 E ee(4) 0.04 E ed(6) 0.08 E ed(6) 0.08 E ed(4) 0.03 E id(7) 0.04 Notes: Figures at the end of the code indicate the sector: 1, Agriculture; 2, Heavy industry; 3, Light industry; 4, Transportation; 5, Construction; 6, Service; 7, Electricity; 8, Coal; 9, Oil, and 10, Natural gas. Table 2 shows that the most sensitive elasticity parameters are generally Eke (elasticity substitution between capital and energy aggregate) and Eee (elasticity substitution between different energy sources). The elasticity parameters in the international trade function (i.e, Eed and Eid) have relatively weak effects on the outputs as shown in the top ten insensitive elasticities in Table 2. When considering sectoral elasticities, the elasticity parameters in heavy industry (Sector 2), service (Sector 6), and natural gas sector (Sector 10) are the most significant contributors listed in the left side of Table 2. That means the elasticities of these three sectors cause the most variations in the outputs. The elasticities in the agriculture, transportation, and oil production sectors (Sectors 1, 4, and 9) have little influence on the output as shown on the right side of Table 2. 4 Conclusions Computable general equilibrium modeling has become an effective technique for evaluating a wide range of policy questions. While uncertainties about the input values in a CGE model may limit the credibility of its conclusions, relatively few applications have explicitly treated the uncertainties. This paper describes formal methods for assessing this type of uncertainties and illustrates its use in a TEDCGE model applied to a carbon tax policy issue. The method relies on building probability density functions of the CGE model output using crude Monte Carlo experiments. In contrast to the traditional opinions, the results indicate that not only can uncertainty in the CGE model be described given full statistical information on the input parameters, but that the uncertainties have important quantitative and qualitative consequences. The input uncertainty can be reduced to some extent through the CGE modeling procedure. The results also indicate that the CGE model results were sensitive to only some of the parameters when the critical parameters for different endogenous variables are varied. The carbon tax level corresponding to a predefined carbon reduction rate in TEDCGE, for example, was quite sensitive to both the capital-energy substitution elasticity and the inter-fuel substitution elasticity in the production sector, while the key parameter affecting the GDP reduction rate was only the inter-fuel substitution elasticity. The results also show that the heavy industry and electricity sectors are the most important sectors affecting the carbon tax level. Acknowledgements The manuscript was mostly prepared during the first author’s stay as a guest researcher at the Energy Project of International Institute for Applied Systems Analysis (IIASA). The authors gratefully acknowledge Dr. Leo Schrattenholzer, Dr. Leonardo Barreto, and Dr. Ji Zou for their valuable comments.
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