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final_demand [2019/06/03 08:16]
jfoure created
final_demand [2020/02/19 12:19] (current)
jschlick Changes of the data section (clarification of the description and sources of elasticity data as well as the update of download links).
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 ====== Final demand ====== ====== Final demand ======
-[{{ ::​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 =====
  
-Government and final consumers are modelled with the same demand function and aggregated in 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 constraintUnless 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 taxHowever, the magnitude ​of the tax revenue losses ​is an interesting information,​ provided in standard result tables.+In each region, ​a representative agent allocates a fixed share of the regional income to savings ​and purchases goods for final consumption with the rest of the incomeThe saving rates follow projections ​by [[MaGE model|EconMap]]The utility function ​of this agent is intra-temporal
  
 +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
 +$$  \eta_{i}^{t} = \frac{\alpha_iI}{P_iC_i^{t}}\left(\frac{P}{P_i}\right)^{\sigma_{i}-1} $$