# Final demand

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 EconMap.

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.

### Data

Utility parameters (minimal consumption level, elasticity of substitution) are calibrated using USDA Commodity and Food Elasticities. 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

All the parameters are calibrated at the same time within an optimization procedure. It consists in minimizing the discrepancies between target and calibrated own-price elasticities subject to the respect of the initial consumption and the income elasticities. The calibration must be done on the consumption per capita. There are usually several solution that minimise the objective (same final own-price elasticities, but different set of parameters), so the initial values 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.

The calibration follows for each country the program $$\min_{\sigma_{i},0\le\bar{c}_{i}\le C_{i}^{t},\alpha_{i}} \sum_{i} \frac{P_{i} C_{i}^{t}}{I}\left(e_{ii}-e_{ii}^{t}\right)^{2}$$ Subject to $$\begin{gather} C_{i}^{t}=Pop_{i}\left(\bar{c}_{i}+\frac{\alpha_{i}\left(I-\gamma\right)}{Pop_{i} P}\left(\frac{P}{P_{i}}\right)^{\sigma_{i}}\right)\\ \eta_{i}^{t} = \frac{\alpha_iI}{P_iC_i^{t}}\left(\frac{P}{P_i}\right)^{\sigma_{i}-1}\\ e_{ii}=-\frac{\alpha_i}{C_i^{t}}\left(\frac{P}{P_i}\right)^{\left(\sigma_{i}-1\right)}\left(C_i^{t}+\sigma_{i}\left(\bar{c}_i-C_i^{t}+\frac{I-\gamma}{P_i}\right)\right) \end{gather}$$ with $e_{ii}^{t}$, $\eta_{i}^{t}$ and $C_{i}^{t}$ the target uncompensated own-price elasticity (this is only for illustration, we use the type of own-price elasticity corresponding to the elasticities sources), the target income elasticity and the target consumption level (from the GTAP database). The parameters are calibrated in consumption per capita.