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final_demand_1.1 [2019/05/29 15:56]
jfoure
final_demand_1.1 [2020/02/19 11:04] (current)
cbellora [Data]
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 ====== Final demand (1.1) ====== ====== Final demand (1.1) ======
-[{{ ::​utility.png?​300|Utility function}}] 
  
-Final consumption is modelled in each region ​through ​a representative agent, whose utility function is intratemporal. A fixed share of the regional income ​is allocated ​to savings, the rest is used to purchase ​final consumption ​goods. The saving ​rate follow projections by [[MaGE model|EconMap]].+===== General approach ===== 
 +In each regiona representative agent allocates a fixed share of the regional income to savings ​and purchases goods for final consumption ​with the rest of the income. The saving ​rates follow projections by [[MaGE model|EconMap]]. The utility function of this agent is intra-temporal
  
-Government ​and final consumers ​are modelled with the same demand function ​and aggregated in a single regional ​representative agent. This agent therefore ​both pays and earns taxes, ​and no public budget constraint has to be taken into account explicitly: this constraint is implicit to meeting ​the representative agent'​s budget constraint. ​Unless ​otherwise indicated (modelling a distorsive replacement tax does not raise any technical problem), ​this implicitly assumes that any decrease in tax revenues (for example as a consequence of a trade liberalisation) is compensated by a non-distorsive replacement tax. However, the magnitude of the tax revenue ​losses is an interesting information,​ provided in standard result tables.+The representative agent gathers government ​and final consumers, meaning that they share the same demand function. Hence, the representative agent both pays and earns taxes. Furthermore, no public budget constraint has to be taken into account explicitly: this constraint is implicit to meet the representative agent'​s budget constraint. ​As a consequence,​ unless ​otherwise indicated (modelling a distorsive replacement tax does not raise any technical problem), any decrease in tax revenues (for example as a consequence of a trade liberalisation) is compensated by a non-distorsive replacement tax. However, the magnitude of the losses ​in tax revenues ​is an interesting information,​ provided in standard result tables.
  
 +===== The LES-CES function =====
 +[{{ ::​utility.png?​300|Utility function}}]
 +
 +We consider that final per capita consumption follows a nesting of CES and LES demand functions. This
 +extension of LES to the slightly more flexible framework of the CES was first proposed by [(:​harvard:​Pollak71)].
 +This LES-CES can be expressed by the following program
 +$$ \max U = \left[\sum_i \alpha_i^{1/​\sigma} (C_i - \bar{c_i}^{(\sigma-1)/​\sigma})\right]^{\sigma/​(\sigma-1)} $$
 +subject to $\sum_i P_iC_i=I$. The consumer demands for each sector $i$ an
 +incompressible level $\bar{c}_{i}$. She allocates the remaining part of her
 +income $I$ according to the CES demand function with elasticity $\sigma$. $C_{i}$
 +represents the consumption from sector $i$ including incompressible
 +consumption. $P_{i}$ is the price of an unitary level of consumption.
 +
 +The LES-CES function inherits from the LES function the possibility to be calibrated on any positive income elasticity. With an income growth, the LES-CES converges to a CES. Own-price elasticities are
 +strongly linked to income elasticities:​ sectors with low income elasticity have
 +a high incompressible consumption. Since most of their consumption is fixed, except
 +for a very high elasticity of substitution in the CES nest, these sectors have a low
 +own-price elasticity. Finally, regularity requires that income be superior to the cost of
 +subsistence $\sum_{i}P_{i}\bar{c}_{i}$.
 ===== Calibration of utility parameters ===== ===== Calibration of utility parameters =====
  
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 ==== Data ==== ==== Data ====
  
-Utility parameters (minimal consumption level, elasticity of substitution) are calibrated using [[https://​www.ers.usda.gov/​data-products/​commodity-and-food-elasticities/​|USDA ​Commodity and Food Elasticities]]. LES-CES can only accommodate positive income elasticities,​ so we change all +Utility parameters (minimal consumption level, elasticity of substitution) are calibrated using elasticities available on the [[https://​www.ers.usda.gov/​data-products/​international-food-consumption-patterns/​|website of the United States Department of Agriculture (USDA)]]. The download page gathers several files: a zip-file including elasticities (of income ​and of prices) for 1996 and Excel files for 2005. More precisely, two datasets are needed: (i) compensated own-price elasticity for broad consumption and (ii) unconditional own-price elasticity for food subcategories. Estimates cover 114 and 144 countries for 1996 and 2005 respectively. For details on how these elasticities ​are obtained, please refer to the [[https://​www.ers.usda.gov/​publications/​pub-details/?​pubid=47430|USDA documentation]].  
-negative income elasticities to a very low value, namely 0.025.+ 
 +LES-CES can only accommodate positive income elasticities,​ so we change all negative income elasticities to a very low value, namely 0.025. 
 + 
  
 ==== Calibration procedure ==== ==== Calibration procedure ====
  
-All the parameters are calibrated at the same time within ​an optimization +All the parameters are calibrated at the same time using an optimization 
-procedure. It consists in minimizing ​the discrepancies between target and +procedure ​that minimizes ​the discrepancies between target and 
-calibrated own-price elasticities subject to the respect of the initial consumption +calibrated own-price elasticitiessubject to the respect of the initial consumption 
-and the income elasticities. The calibration ​must be done on the consumption per +and the income elasticities. The calibration ​is done on the consumption per 
-capita. There are usually several ​solution ​that minimise the objective (same final +capita. There are usually several ​solutions ​that minimise the objective (same final 
-own-price elasticities,​ but different set of parameters), ​so the initial values +own-price elasticities,​ but different set of parameters), ​depending on the initial values. These multiple solutions correspond to the same
-matter a lot for the calibration. These multiple solutions correspond to the same+
 initial behavior but to different behavior after a change in prices or income. initial behavior but to different behavior after a change in prices or income.
  
- +For each country, the calibration that uses as target the own price elasticity follows ​the program 
-The calibration follows for each country the program +$$\min_{\sigma_{i},​0 \le \bar{c}_{i} \le C_{i}^{t}, \alpha_{i}} \sum_{i} \frac{P_{i} C_{i}^{t}}{I}\left(e_{ii}-e_{ii}^{t}\right)^{2}$$
-$$\min_{\sigma_{i},​0\le\bar{c}_{i}\le C_{i}^{t},​\alpha_{i}} \sum_{i} \frac{P_{i} C_{i}^{t}}{I}\left(e_{ii}-e_{ii}^{t}\right)^{2}$$+
 Subject to Subject to
 $$ $$
 \begin{gather} \begin{gather}
-  C_{i}^{t}=Pop_{i}\left(\bar{c}_{i}+\frac{\alpha_{i}\left(I-\gamma\right)}{Pop_{i} +  C_{i}^{t}=Pop_{i} \left[\bar{c}_{i}+\frac{\alpha_{i}\left(I-\gamma\right)}{Pop_{i} 
-    P}\left(\frac{P}{P_{i}}\right)^{\sigma_{i}}\right)\\ +    P}\left(\frac{P}{P_{i}}\right)^{\sigma_{i}}\right]\\ 
-  \eta_{i}^{t} = +  e_{ii}=-\frac{\alpha_i}{C_i^{t}}\left(\frac{P}{P_i}\right)^{\left(\sigma_{i}-1\right)}\left[C_i^{t}+\sigma_{i}\left(\bar{c}_i-C_i^{t}+\frac{I-\gamma}{P_i}\right)\right]
-  \frac{\alpha_iI}{P_iC_i^{t}}\left(\frac{P}{P_i}\right)^{\sigma_{i}-1}\\ +
-  e_{ii}=-\frac{\alpha_i}{C_i^{t}}\left(\frac{P}{P_i}\right)^{\left(\sigma_{i}-1\right)}\left(C_i^{t}+\sigma_{i}\left(\bar{c}_i-C_i^{t}+\frac{I-\gamma}{P_i}\right)\right)+
 \end{gather} \end{gather}
 $$ $$
-with $e_{ii}^{t}$, $\eta_{i}^{t}$ and $C_{i}^{t}$ ​the target uncompensated+with $e_{ii}^{t}$ ​is the target uncompensated
 own-price elasticity (this is only for illustration,​ we use the type of own-price elasticity (this is only for illustration,​ we use the type of
-own-price elasticity corresponding to the elasticities sources), the target +own-price elasticity corresponding to the elasticities sources), ​$C_{i}^{t}$ is the target 
-income elasticity and the target ​consumption level (from the GTAP database). The +consumption level (from the GTAP database), $\gamma = \sum_i P_i\bar{c}_i$ and $P=\left(\sum_i \alpha_iP_i^{1-\sigma}\right)^{\frac{1}{1-\sigma}}$.
-parameters are calibrated in consumption per capita.+
  
 +According to the source availability,​ the calibration is also done targeting the 
 +income elasticity $\eta_{i}$, with
 +$$  \eta_{i}^{t} = \frac{\alpha_iI}{P_iC_i^{t}}\left(\frac{P}{P_i}\right)^{\sigma_{i}-1} $$