Equations of the model (Outdated)

The i and j indices refer to sectors, r and s refer to regions, t to periods. Superscripts for prices P refer to the related variable.

U(s) is the subset of countries in the same development level as region s and V(s) is the subset of countries with a different level of development.

Agri(i) is the subset of sectors from agriculture.

$i_{\mathit{TrT}}$ refers to transport sectors and $r_{\mathit{EU}}$ refers to the European Union regions. The reference year is indexed with $t_0$.

$$ \begin{array}{ll} \sigma_{\mathit{VA}_j}\ \sigma_{\mathit{CAP}_j}\ \sigma_C\ \sigma_{\mathit{IC}}\ \sigma_{\mathit{KG}} &\\ \sigma_{\mathit{GEO}_i}\ \sigma_{\mathit{ARM}_i}\ \sigma_{\mathit{IMP}_i}\ \sigma_{\mathit{VAR}_i} & Substitution \ elasticities \ of \ factors \ and \ goods \ demand\\ \mathit{cmin}_{i,r}&Minimal \ consumption \ of \ good \ i \ in \ the \ final \ demand \ of \ region \ r\\ \mathit{epa}_r&Saving \ rate \ in \ region \ r\\ \mu_{i,r,s}& Transport \ demand \ per \ volume \ of \ good\\ \theta_r& Value \ share \ of \ region \ r \ transport \ sector \ in \ the \ world \ production \ of \ transport\\ \mathit{DD}_{i,r,s,t}& Ad-valorem \ tariff \ rate \ applied \ by \ regions \ s \ on \ its \ imports \ from \ region \ r\\ \mathit{MaxExpSub}_{i,r,t}&Maximum \ level \ of \ subsidized \ exports \ authorized \ by \ the \ WTO\\ \mathit{taxp}_{i,r} \mathit{taxcc}_{i,s}\ \mathit{taxicc}_{i,s}\ \mathit{taxkgc}_{i,s}&Tax \ rates \ applied \ on \ production, \ final \ consumption, \ intermediate \ consumption \ and \ capital \ good\\ \mathit{taxAMF}_{i,r,s}&Export \ tax \ rate \ equivalent \ to \ the \ Multifibre \ Arrangement\\ \mathit{TsubK}_{i,r}&Subsidy \ rate \ on \ capital\\ \mathit{TsubTE}_{i,r}&Subsidy \ rate \ on \ land\\ \mathit{cf}_{j,r}&Fixed \ cost \ per \ firm, \ in \ units \ of \ output \(imperfectly \ competitive \ sectors)\\ \mathit{mmoy}_{i,r}&Mark-up \ average\\ \mathit{Quota}_{i,r,t}&Maximum \ production \ in \ sectors \ where \ quotas \ hold\\ \alpha &Elasticity \ of \ investment \ to \ capital \ return \ rate\\ \gamma_{i,r}^L\ \gamma_{i,r}^Q\ \gamma_{i,r}^\mathit{TE}\ \gamma_{i,r}^\mathit{RN}&Value \ share \ of \ factors \ in \ value \ added \ (Cobb-Douglas)\\ \delta&Depreciation \ of \ capital\\ \rho_{r,t}&Population \ growth \ rate \ of \ region \ r \ (World \ Bank \ data)\\ a_{\mathit{XXX}}& Various \ share \ and \ scale \ coefficients \ in \ CES \ or \ Cobb-Douglas \ functions\\ \mathit{PGF}_{r,t}& Total \ factor \ productivity \end{array} $$

Production

$$ \begin{array}{ll} \mathit{Y}_{i,r,t} & Output \ of \ sector \ i \ firms\\ \mathit{VA}_{i,r,t}& Value \ added\\ \mathit{CNTER}_{i,r,t}& Aggregate \ intermediate \ consumption\\ \end{array} $$

Factors

$$ \begin{array}{ll} \mathit{Q}_{i,r,t}& Aggregate \ of \ human \ capital \ and \ physical \ capital\\ \mathit{L}_{i,r,t}& Unskilled \ labour\\ \mathit{L}^{\mathit{Agri}_{i,r,t}}& Total \ Unskilled \ labour \ in \ agriculture\\ \mathit{L}^{\mathit{notAgri}}_{i,r,t}& Total \ Unskilled \ labour \ in \ sectors \ other \ than \ agriculture\\ \mathit{TE}_{i,r,t}& Land\\ \mathit{RN}_{i,r,t}&Natural \ resources\\ \mathit{H}_{i,r,t}&Skilled \ labour\\ \mathit{K}_{i,r,s,t}&Capital \ stock \ from \ region \ r \ to \ region \ s \ in \ sector \ i\\ \mathit{KTOT}_{i,r,t}&Total \ capital \ stock \ in \ sector \ i \ and \ region \ r\\ \overline{\mathit{L}}_{r,t}&Total \ supply \ of \ unskilled \ labour\\ \overline{\mathit{TE}}_{r,t}&Total \ supply \ of \ land\\ \overline{\mathit{H}}_{r,t}&Total \ supply \ of \ skilled \ labour\\ \overline{\mathit{K}}_{r,t}&Total \ supply \ of \ capital\\ \end{array} $$

Demand

$$ \begin{array}{ll} \mathit{BUDC}_{r,t}&Budget \ allocated \ to \ consumption\\ \mathit{UT}_{r,t}&Utility\\ \mathit{P}_{r,t}&Price \ of \ utility\\ \mathit{C}_{i,r,t}&Aggregated \ consumption\\ \mathit{IC}_{i,j,r,t}&Intermediate \ consumption \ of \ good \ i \ used \ in \ the \ production \ of \ sector \ j\\ \mathit{INVTOT}_{r,t}&Total \ investment \ in \ region \ r\\ \mathit{INV}_{i,r,s,t}&Investment \ from \ region \ r \ to \ sector \ i \ in \ region \ s\\ \mathit{B}_{r,t}&Investment \ scale \ coefficient\\ \mathit{KG}_{i,r,t}&Capital \ good \ demand \ of \ sector \ i \ in \ region \ r\\ \mathit{DEMTOT}_{i,r,t}&Total \ demand\\ \mathit{DEMU}_{i,r,t}&Total \ demand, \ in \ region \ r, \ of \ good \ originating \ from \ regions \ with \ the \ same \ development \ level \ as \ region \ r \ (including \ local \ demand \ in \ region \ r)\\ \mathit{DEMV}_{i,r,t}& Total \ demand, \ in \ region \ r, \ of \ good \ originating \ from \ regions \ with \ a \ different \ development \ level \ from \ region \ r\\ \mathit{D}_{i,r,t}&Domestic \ demand \ of \ good \ i\\ \mathit{DVAR}_{i,r,t}&Domestic \ demand \ of \ good \ i \ produced \ by \ each \ firm \ of \ region r\\ M_{i,r,t}&Total \ demand, \ in \ region \ r, \ of \ good \ i \ originating \ from \ regions \ with \ the \ same \ development \ level \ as \ region \ r \ other \ than \ region \ r\\ \mathit{DEM}_{i,r,s,t}&Demand, \ in \ region \ s, \ of \ good \ i \ originating \ from \ region \ r\\ \mathit{DEMVAR}_{i,r,s,t}&Demand \ of \ good \ i \ produced \ by \ each \ firm \ of \ region \ r\\ \end{array} $$

Transportation

$$ \begin{array}{ll} \mathit{TRADE}_{i,r,s,t}&Exports \ to \ region \ s \ of \ industry \ i \ in \ region \ r\\ \mathit{TR}_{i,r,s,t}&Transport \ demand\\ \mathit{MONDTR}_t&Transport \ aggregate\\ P^T_t&Transport \ of \ commodities \ price\\ \mathit{TRM}_{i,r,t}&Supply \ of \ international \ transportation \ sector \ i \ in \ region \ r\\ \end{array} $$

Monopolistic competition

$$ \begin{array}{ll} \mathit{EP}_{i,r,s,t}&Perceived \ price \ elasticity \ of \ total \ demand\\ \mathit{EPD}_{i,r,t}&Perceived \ price \ elasticity \ of \ domestic \ demand\\ \mathit{NB}_{i,r,t}&Number \ of \ varieties \ in \ imperfectly \ competitive \ sectors\\ \mathit{SDU}_{i,s,t}&Market \ share \ of \ domestic \ demand \ in \ demand \ of \ regions \ with \ the \ same \ level \ of \ development \ as \ region r\\ \mathit{SDT}_{i,s,t}&Market \ share \ of \ domestic \ demand \ in \ total \ demand\\ \mathit{SE}_{i,r,s,t}&Market \ share \ of \ imports \ from \ region \ r \ in \ imports \ of \ region \ s \ originating \ from \ regions \ with \ the \ same \ level \ of \ development\\ \mathit{SU}_{i,r,s,t}& Market \ share \ of \ imports \ from \ region \ r \ in \ demand \ of \ region \ s \ for \ goods \ from \ regions \ with \ the \ same \ level \ of \ development\\ \mathit{SV}_{i,r,s,t}&Market \ share \ of \ imports \ from \ region \ r \ in \ imports \ of \ region \ s \ originating \ from \ regions \ with \ a \ different \ level \ of \ development\\ \mathit{ST}_{i,r,s,t}&Market \ share \ of \ imports \ from \ region \ r \ in \ demand \ of \ region \ s\\ \end{array} $$

Tax revenue

$$ \begin{array}{ll} \mathit{RECPROD}_{i,r,t}&Revenue \ of \ production \ tax\\ \mathit{RECDD}_{i,r,t}&Revenue \ of \ tariff\\ \mathit{RECCONS}_{i,r,t}&Revenue \ of \ consumption \ tax\\ \mathit{RECEXP}_{i,r,t}&Revenue \ of \ exports \ tax\\ \mathit{RECTAX}_{r,t}&Total \ tax \ revenue\\ \mathit{RQUOTA}_{i,r,s,t}&Implicit \ transfers \ due \ to \ quotas\\ \mathit{REV}_{r,t}&Regional \ revenue\\ \mathit{SOLD}_{r,t}&Current \ account \ balance\\ \mathit{PIBMVAL}_t&Total \ GDP \ in \ value\\ \mathit{GDPVOL}_{r,t}&Regional \ GDP\\ \end{array} $$

Prices and taxes

$$ \begin{array}{ll} P^\mathit{XXX}&Generic \ notation \ to \ indicate \ the \ price \ of \ the \ variable \ XXX\\ P^\mathit{CIF}_{i,r,s,t}&CIF \ price\\ P^\mathit{Int}_{i,t}&Intervention \ price \ (European \ Union \ only)\\ W^{\overline{K}}_{r,t}&Capital \ return \ rate \ in \ region \ r\\ W^{K}_{i,r,t}&Capital \ return \ paid \ to \ the \ investor\\ W^{\overline{\mathit{TE}}}_{r,t}&Land \ return \ rate \ in \ region \ r\\ W^{\mathit{TE}}_{i,r,t}&Land \ return \ rate \ paid \ to \ the \ owner\\ \mathit{TAXEXP}_{i,r,s,t}&Export \ tax \ rate\\ \mathit{TAXREF}_{i,r,s,t}&Auxiliary \ variable \ to \ adjust \ TAXMOY \ to \ its \ proper \ level \ while \ keeping \ unchanged \ the \ distribution \ across \ destinations\\ \mathit{TAXMOY}_{i,r,t}&Average \ export \ tax \ rate \ across \ destinations\\ \end{array} $$

Supply

Determination of supply results from the following optimization programs:

Leontieff relation between value added and intermediate consumption

  • Imperfect competition

$$ \quad \min\mathit{NB}_{i,r,t} P^Y_{i,r,t} (Y_{i,r,t} + \mathit{cf}_{i,r}) = P^{\mathit{VA}}_{i,r,t} \mathit{VA}_{i,r,t} + P^{\mathit{CNTER}}_{i,r,t} \mathit{CNTER_{i,r,t}}\\ $$

s.t.

$$ \quad \mathit{NB}_{i,r,t} (Y_{i,r,t} + \mathit{cf}_{i,r}) = a^{\mathit{VA}}_{i,r} \mathit{VA}_{i,r,t} = a^{\mathit{CNTER}}_{i,r} \mathit{CNTER}_{i,r,t} $$

  • Perfect competition

$$ \quad \min P^Y_{i,r,t}Y_{i,r,t} = P^{\mathit{VA}}_{i,r,t}\mathit{VA}_{i,r,t} + P^{\mathit{CNTER}}_{i,r,t}\mathit{CNTER}_{i,r,t} + P^{\mathit{Quota}}_{i,r,t}\mathit{Quota}_{i,r,t}\\ $$

s.t.

$$ \quad Y_{i,r,t} = a^{\mathit{VA}}_{i,r} \mathit{VA}_{i,r,t} = a^{\mathit{CNTER}}_{i,r} \mathit{CNTER}_{i,r,t} $$

For sectors where quotas hold (perfect competition only): $$ Y_{i,r,t} = \mathit{Quota}_{i,r,t} $$

Factor demand

  • Value-added is given by

$$ \quad \min P^{\mathit{VA}}_{i,r,t}\mathit{VA}_{i,r,t} = P^L_{i,r,t} L_{i,r,t} + P^Q_{i,r,t} Q_{i,r,t} + P^{\mathit{TE}}_{i,r,t}\mathit{TE}_{i,r,t} + P^{\mathit{RN}}_{i,r,t} \mathit{RN}_{i,r,t}\\ $$

s.t. $$ \quad \left(\frac{\mathit{VA}_{i,r,t}}{\mathit{PGF}_{r,t}}\right)^{1-\frac{1}{\sigma_{\mathit{VA}_i}}} = a^L_{i,r} L_{i,r,t}^{1-\frac{1}{\sigma_{\mathit{VA}_i}}}+a^Q_{i,r} Q_{i,r,t}^{1-\frac{1}{\sigma_{\mathit{VA}_i}}}+a^{\mathit{RN}}_{i,r} \mathit{RN}_{i,r,t}^{1-\frac{1}{\sigma_{\mathit{VA}_i}}}+a^{\mathit{TE}}_{i,r} \mathit{TE}_{i,r,t}^{1-\frac{1}{\sigma_{\mathit{VA}_i}}}\\ \quad \quad $$ (CES option)

or s.t. $$ \quad \mathit{VA}_{i,r,t}=A_{i,r}\mathit{PGF}_{r,t}{L_{i,r,t}}^{\gamma^L_{i,r}}{Q_{i,r,t}}^{\gamma^Q_{i,r}}{\mathit{TE}_{i,r,t}}^{\gamma^{\mathit{TE}}_{i,r}}{\mathit{RN}_{i,r,t}}^{\gamma^{\mathit{RN}}_{i,r}} \quad $$ (Cobb-Douglas option)

  • in which capital-skilled labour bundle:

$$ \quad \min P^Q_{i,r,t}Q_{i,r,t} = P^K_{i,r,t}\mathit{KTOT}_{i,r,t}+P^H_{i,r,t}H_{i,r,t}\\ $$

s.t. $$ \quad Q_{i,r,t}^{1-\frac{1}{\sigma_{\mathit{CAP}_i}}} = a^K_{i,r} \mathit{KTOT}_{i,r,t}^{1-\frac{1}{\sigma_{\mathit{CAP}_i}}}+a^H_{i,r} H_{i,r,t}^{1-\frac{1}{\sigma_{\mathit{CAP}_i}}} $$

The capital stock in region s is described by: $$ \quad \mathit{KTOT}_{i,s,t}=\sum_r K_{i,r,s,t} $$

Comment: in this model, production quotas have been introduced. For the associated sectors, production is equal to the quota and an additional income, equal to $P^{\mathit{Quota}}_{i,r,t}\mathit{Quota}_{i,r,t}$, is drawn from the quota.

Demand

Determination of demand results from the following optimization programs:

LES-CES (first stage)

$$ \quad \min P_{r,t} \mathit{UT}_{r,t} = \sum_i P^C_{i,r,t}(C_{i,r,t}-\mathit{cmin}_{i,r})\\ $$

s.t. $$ \quad {\mathit{UT}_{r,t}}^{1-\frac{1}{\sigma_C}} = \sum_i a^C_{i,r}(C_{i,r,t}-\mathit{cmin}_{i,r})^{1-\frac{1}{\sigma_{C}}}\\ $$

Budget equations

$$ \begin{array}{ll} \quad \mathit{BUDC}_{r,t}&=\sum_i P^C_{i,r,t} C_{i,r,t}\\ \quad P^C_{i,r,t} &= P^{\mathit{DEMTOT}}_{i,r,t}(1+\mathit{taxcc}_{i,r})\\ \quad P^{\mathit{KG}}_{i,r,t} &= P^{\mathit{DEMTOT}}_{i,r,t}(1+\mathit{taxkgc}_{i,r})\\ \quad \mathit{DEMTOT}_{i,r,t}&=C_{i,r,t}+\sum_j\mathit{IC}_{i,j,r,t}+\mathit{KG}_{i,r,t} \end{array} $$

Groups of regions (second stage)

$$ \quad \min P^{\mathit{DEMTOT}}_{i,r,t}\mathit{DEMTOT}_{i,r,t} =P^{\mathit{DEMU}}_{i,r,t}\mathit{DEMU}_{i,r,t}+P^{\mathit{DEMV}}_{i,r,t}\mathit{DEMV}_{i,r,t}\\ $$

s.t. $$ \quad \mathit{DEMTOT}_{i,r,t}^{1-\frac{1}{\sigma_{\mathit{GEO}_i}}} = a^{\mathit{DEMU}}_{i,r} \mathit{DEMU}_{i,r,t}^{1-\frac{1}{\sigma_{\mathit{GEO}_i}}}+a^{\mathit{DEMV}}_{i,r} \mathit{DEMV}_{i,r,t}^{1-\frac{1}{\sigma_{\mathit{GEO}_i}}} $$

Armington (third stage)

$$ \quad \min P^{\mathit{DEMU}}_{i,r,t}\mathit{DEMU}_{i,r,t} =P^D_{i,r,t}D_{i,r,t}+P^M_{i,r,t}M_{i,r,t}\\ $$

s.t. $$ \quad \mathit{DEMU}_{i,r,t}^{1-\frac{1}{\sigma_{\mathit{ARM}_i}}} = a^{\mathit{DEM}}_{i,r} \mathit{D}_{i,r,t}^{1-\frac{1}{\sigma_{\mathit{ARM}_i}}}+a^M_{i,r} M_{i,r,t}^{1-\frac{1}{\sigma_{\mathit{ARM}_i}}} $$

Regions (fourth stage)

  • For foreign regions with the same level of development:

$$ \quad \min P^M_{i,s,t}M_{i,s,t} = \sum_{r\in U(s)} P^{\mathit{DEM}}_{i,r,s,t}\mathit{DEM}_{i,r,s,t}\\ $$

s.t. $$ \quad M_{i,s,t}^{1-\frac{1}{\sigma_{\mathit{IMP}_i}}} = \sum_{r\in U(s)} a^{\mathit{IMP}}_{i,r,s}\mathit{DEM}_{i,r,s,t}^{1-\frac{1}{\sigma_{\mathit{IMP}_i}}} $$

* For foreign regions with different levels of development:

$$ \min P^{\mathit{DEMV}}_{i,s,t}\mathit{DEMV}_{i,s,t} = \sum_{r\in V(s)} P^{\mathit{DEM}}_{i,r,s,t}\mathit{DEM}_{i,r,s,t}\\ $$

s.t. $$ \quad \mathit{DEMV}_{i,s,t}^{1-\frac{1}{\sigma_{\mathit{IMP}_i}}} = \sum_{r\in V(s)} a^{\mathit{IMP}}_{i,r,s}\mathit{DEM}_{i,r,s,t}^{1-\frac{1}{\sigma_{\mathit{IMP}_i}}} $$

Varieties (fifth stage)

$$ \begin{array}{ll} \quad \mathit{DEMVAR}_{i,r,s,t} &= \mathit{DEM}_{i,r,s,t}\mathit{NB}_{i,r,t}^{1-\frac{1}{\sigma_{\mathit{VAR}_i}}}\\ \quad P^{\mathit{DEM}}_{i,r,s,t} &= P^{\mathit{DEMVAR}}_{i,r,s,t}\mathit{NB}_{i,r,t}^{1-\frac{1}{\sigma_{\mathit{VAR}_i}}}\\ \quad \mathit{DVAR}_{i,s,t} &= D_{i,s,t}\mathit{NB}_{i,s,t}^{1-\frac{1}{\sigma_{\mathit{VAR}_i}}}\\ \quad P^D_{i,s,t} &= P^{\mathit{DVAR}}_{i,r,t}\mathit{NB}_{i,s,t}^{1-\frac{1}{\sigma_{\mathit{VAR}_i}}} \end{array} $$

Intermediate consumption

$$ \quad P^{\mathit{IC}}_{i,j,r,t} = P^{\mathit{DEMTOT}}_{i,r,t}(1+\mathit{taxicc}_{i,j,r}) $$

$$ \quad \min P^{\mathit{CNTER}}_{j,r,t}\mathit{CNTER}_{j,r,t}= \sum_i P^{\mathit{IC}}_{i,j,r,t}\mathit{IC}_{i,j,r,t}\\ $$

s.t. $$ \quad \mathit{CNTER}_{j,r,t}^{1-\frac{1}{\sigma_{\mathit{IC}}}} = \sum_i a^{\mathit{IC}}_{i,j,r} \mathit{IC}_{i,j,r,t}^{1-\frac{1}{\sigma_{\mathit{IC}}}} $$

Capital good

$$ \quad \min P^{\mathit{INVTOT}}_{r,t}\mathit{INVTOT}_{r,t} = \sum_i P^{\mathit{KG}}_{i,r,t}\mathit{KG}_{i,r,t}\\ $$

s.t. $$ \quad \mathit{INVTOT}_{r,t}^{1-\frac{1}{\sigma_{\mathit{KG}}}} = \sum_i a^{\mathit{KG}}_{i,r} \mathit{KG}_{i,r,t}^{1-\frac{1}{\sigma_{\mathit{KG}}}} $$

Commodity market equilibrium

  • Imperfect competition

$$ \begin{array}{ll} \quad Y_{i,r,t}=\mathit{DVAR}_{i,r,t}+\sum_s\mathit{DEMVAR}_{i,r,s,t}\\ \quad \mathit{TRADE}_{i,r,s,t} =\mathit{NB}_{i,r,t}\mathit{DEMVAR}_{i,r,s,t}\\ \end{array} $$

  • Perfect competition

$$ \begin{array}{ll} \quad Y_{i,r,t}&=D_{i,r,t}+\sum_s\mathit{DEM}_{i,r,s,t}\quad (i\notin\mathit{TrT})\\ \quad Y_{i_{\mathit{TrT}},r,t}&=D_{i_{\mathit{TrT}},r,t}+\sum_s\mathit{DEM}_{i_{\mathit{TrT}},r,s,t}+\mathit{TRM}_{i_{\mathit{TrT}},r,t}\\ \quad \mathit{TRADE}_{i,r,s,t}&=\mathit{DEM}_{i,r,s,t} \end{array} $$

Transport sector

Transport demand

$$ \begin{array}{ll} \quad \mathit{TR}_{i,r,s,t}&=\mu_{i,r,s}\mathit{TRADE}_{i,r,s,t}\\ \quad \mathit{MONDTR}_t&=\sum_{i,r,s}\mathit{TR}_{i,r,s,t}\\ \end{array} $$

Transport supply

$$ \begin{array}{ll} \quad P^Y_{i_\mathit{TrT},r,t}(1+\mathit{taxp}_{i_{\mathit{TrT}},r})\mathit{TRM}_{i_{\mathit{TrT}},r,t}&=\theta_{i_{\mathit{TrT}},r} P^T_t\mathit{MONDTR}_t\\ \quad \mathit{MONDTR}_t&=a^T\prod_r{\mathit{TRM}_{i_{\mathit{TrT}},r,t}}^{\theta_{i_{\mathit{TrT}},r}} \end{array} $$

Factor market

Labour market

  • Developed countries: labour allocation between agricultural and non agricultural sectors

$$ \begin{array}{ll} L^{Agri}_{r,t}&=b_r^{L^{Agri}}\overline{L}_{r,t} \left(\frac{P^{L^{Agri}}_{r,t}}{P^{\overline{L}}_{r,t}}\right)^{\sigma_L}\\ L^{notAgri}_{r,t}&=b_r^{L^{notAgri}}\overline{L}_{r,t} \left(\frac{P^{L^{notAgri}}_{r,t}}{P^{\overline{L}}_{r,t}}\right)^{\sigma_L} \end{array} $$

  • Developing countries: dual labour market

$$ \begin{array}{ll} P^{L^{notAgri}}_{r,t} &= P^{L^{notAgri}}_{r,t,Ref} \prod_i\left(\frac{P^C_{i,r,t}}{P^C_{i,r,{Ref}}}\right)^\frac{P^C_{i,r,{t_0}}C_{i,r,{t_0}}}{\sum\limits _j P^C_{j,r,{t_0}}C_{j,r,{t_0}}}\\ L^{Agri}_{r,t}&=L^{Agri}_{r,t,Ref} \end{array} $$

where $L^{notAgri}_{r,t,Ref}$ and $L^{Agri}_{r,t,Ref}$ are the baseline Ref labour supply exogenously calculated from migration flows in FAO data.

$P^{L^{notAgri}}_{r,t,Ref}$ is computed endogenously from $L^{notAgri}_{r,t,Ref}$ in the baseline.

Labour market (both cases)

$$ P^{\overline{L}}_{r,t}\overline{L}_{r,t}=P^{L^{Agri}}_{r,t}L^{Agri}_{r,t}+P^{L^{notAgri}}_{r,t}L^{notAgri}_{r,t} $$

Land market

$$ \quad W^{\mathit{TE}}_{i,r,t}=P^{\mathit{TE}}_{r,t}+P_{r,t}\mathit{TsubTE}_{i,r,t} $$

  • Land supply

$$ \begin{array}{ll} \quad W^{\overline{\mathit{TE}}}_{r,t}\overline{\mathit{TE}}_{r,t}&=\sum_i W^{\mathit{TE}}_{i,r,t} \mathit{TE}_{i,r,t}\\ \quad \overline{\mathit{TE}}_{r,t}&=\overline{\mathit{TE}}_{r,{t_0}} \left({W^{\overline{\mathit{TE}}}_{r,t}}\right)^{\sigma_{\overline{\mathit{TE}}}}\quad (\mathrm{Nb:\: }W^{\overline{\mathit{TE}}}_{r,{t_0}}=1)\\ \end{array} $$

  • Land allocation

$$ \quad \mathit{TE}_{i,r,t} =b^{\mathit{TE}}_{i,r} \overline{\mathit{TE}}_{r,t}\left(\frac{W^{\mathit{TE}}_{i,r,t}}{W^{\overline{\mathit{TE}}}_{r,t}}\right)^{\sigma_{\mathit{TE}}} $$

Full use of factor endowments

$$ \begin{array}{ll} \quad L^{\mathit{Agri}}_{r,t}&=\sum_{j\in Agri(j)} L_{j,r,t}\\ \quad L^{\mathit{notAgri}}_{r,t}&=\sum_{j\notin Agri(j)} L_{j,r,t}\\ \quad \overline{\mathit{TE}}_{r,t}&=\sum_j \mathit{TE}_{j,r,t}\\ \quad \overline{\mathit{H}}_{r,t}&=\sum_j H_{j,r,t} \end{array} $$

Comments:

  • In comparison to the standard model, the agricultural version distinguishes between two types of unskilled labour: agricultural labour and non agricultural labour. A partial mobility between these two types of labour is allowed through a Constant Elasticity of Transformation supply function. Within each category, labour is perfectly mobile.
  • A duality of labour has been assumed in developing countries: an efficiency wage scheme determines the level of wages in non agricultural sectors and the corresponding labour demand, while labour demand in agricultural sectors is exogenous. The efficiency wage is set such that the purchasing power of non agricultural wages remains unchanged after the shock.

Factor mobility

$$ \begin{array}{ll} \quad P^L_{i,r,t}&=P_{r,t}^{L^\mathit{Agri\quad}}\quad (i\in \mathit{Agri(i)})\\ \quad P^L_{i,r,t}&=P_{r,t}^{L^\mathit{notAgri}}\quad (i\notin \mathit{Agri(i)})\\ \quad P^{\mathit{TE}}_{i,r,t}&=P_{r,t}^{\overline{\mathit{TE}}}\\ \quad P^H_{i,r,t}&=P_{r,t}^{\overline{H}} \end{array} $$

Revenues

Firms maximisation of profit

(Imperfect competition only)

$$ \quad 0=P^Y_{i,r,t} \left(\mathit{NB}_{i,r,t} \sum_s \frac{\mathit{DEMVAR}_{i,r,s,t}}{1+EP_{i,r,s,t}}+\frac{\mathit{NB}_{i,r,t}\mathit{DVAR}_{i,r,t}}{1+\mathit{EPD}_{i,r,t}}\right) -(P^\mathit{VA}_{i,r,t}\mathit{VA}_{i,r,t}+P^\mathit{CNTER}_{i,r,t}\mathit{CNTER}_{i,r,t}) $$

Comment: this corresponds to the zero profit condition allowing to compute the number of firms.

Tax revenue

  • Imperfectly competitive sectors

$$ \begin{array}{ll} \quad \mathit{RECPROD}_{i,r,t}&=\mathit{taxp}_{i,r}P^Y_{i,r,t} \left(\mathit{NB}_{i,r,t} \sum_s \frac{\mathit{DEMVAR}_{i,r,s,t}}{1+EP_{i,r,s,t}}+\frac{\mathit{NB}_{i,r,t}\mathit{DVAR}_{i,r,t}}{1+\mathit{EPD}_{i,r,t}}\right)\\ \quad \mathit{RECEXP}_{i,r,t}&=(1+\mathit{taxp}_{i,r})P^Y_{i,r,t} \mathit{NB}_{i,r,t} * \sum_s (\mathit{TAXEXP}_{i,r,s,t}+\mathit{taxAMF}_{i,r,s,t})\frac{\mathit{DEMVAR}_{i,r,s,t}}{1+EP_{i,r,s,t}}\\ \end{array} $$

  • Perfectly competitive sectors

$$ \begin{array}{ll} \quad \mathit{RECPROD}_{i,r,t}&=\mathit{taxp}_{i,r}P^Y_{i,r,t} Y_{i,r,t}\\ \quad \mathit{RECEXP}_{i,r,t}&=(1+\mathit{taxp}_{i,r})P^Y_{i,r,t} * \sum_s (\mathit{TAXEXP}_{i,r,s,t}+\mathit{taxAMF}_{i,r,s,t})\mathit{TRADE}_{i,r,s,t}\\ \end{array} $$

  • For both sectors:

$$ \begin{array}{ll} \quad \mathit{RECDD}_{i,r,t}&=\sum_r \mathit{DD}_{i,r,s,t} P^{\mathit{CIF}}_{i,r,s,t} \mathit{TRADE}_{i,r,s,t}\\ \quad \mathit{RQUOTA}_{r,s,t}&=\sum_{i\in\mathit{TQUOTAO}}\mathit{TQUOTA}_{i,r,s,t}P^{\mathit{CIF}}_{i,r,s,t} \quad \mathit{TRADE}_{i,r,s,t}\\ \quad \mathit{RECCONS}_{i,s,t}&=P^\mathit{DEMTOT}_{i,s,t}(\mathit{taxcc}_{i,s}C_{i,s,t}+\mathit{taxkgc_{i,s}}\mathit{KG}_{i,s,t} +\sum_j\mathit{taxicc}_{i,j,s,t}\mathit{IC}_{i,j,s,t})\\ \quad \mathit{RECTAX}_{r,t}&=\sum_i \mathit{RECPROD}_{i,r,t}+\mathit{RECEXP}_{i,r,t} +\mathit{RECDD}_{i,r,t}+\mathit{RECCONS}_{i,r,t}\\ \end{array} $$

Savings

$$ \quad \mathit{BUDC}_{r,t}=(1-\mathit{epa}_r)\mathit{REV}_{r,t}\\ $$

Prices definition

Sale price

  • Imperfect competition

$$ \begin{array}{ll} \quad P^{\mathit{DEMVAR}}_{i,r,s,t}&=P^{\mathit{CIF}}_{i,r,s,t}(1+\mathit{DD}_{i,r,s,t})\\ \quad P^{\mathit{DVAR}}_{i,r,t}&=\frac{P^Y_{i,r,t}(1+\mathit{taxp}_{i,r})}{1+\mathit{EPD}_{i,r,t}}\\ \end{array} $$

  • Perfect competition

$$ \begin{array}{ll} \quad P^{\mathit{DEM}}_{i,r,s,t}&=P^{\mathit{CIF}}_{i,r,s,t}(1+\mathit{DD}_{i,r,s,t})\\ \quad P^\mathit{D}_{i,r,t}&=P^Y_{i,r,t}(1+\mathit{taxp}_{i,r})\\ \end{array} $$

CIF price

  • Imperfect competition

$$ \quad P^{\mathit{CIF}}_{i,r,s,t}=(1+\mathit{taxp}_{i,r})(1+\mathit{TAXEXP}_{i,r,s,t}+\mathit{taxAMF}_{i,r,s,t})\frac{P^Y_{i,r,t}}{1+\mathit{EP}_{i,r,s,t}}+\mu_{i,r,s}P^T_t $$

  • Perfect competition

$$ \quad P^{\mathit{CIF}}_{i,r,s,t}=(1+\mathit{taxp}_{i,r})(1+\mathit{TAXEXP}_{i,r,s,t}+\mathit{taxAMF}_{i,r,s,t}) P^Y_{i,r,t}+\mu_{i,r,s}P^T_t $$

Imperfect competition

Determination of market shares

$$ \begin{array}{ll} \quad \mathit{SDU}_{i,s,t}&=\frac{P^\mathit{D}_{i,s,t}\mathit{D}_{i,s,t}}{P^\mathit{DEMU}_{i,s,t}\mathit{DEMU}_{i,s,t}}\\ \quad \mathit{SDT}_{i,s,t}&=\frac{P^\mathit{D}_{i,s,t}\mathit{D}_{i,s,t}}{P^\mathit{DEMTOT}_{i,s,t}\mathit{DEMTOT}_{i,s,t}}\\ \quad \mathit{SE}_{i,r,s,t}&=\frac{P^\mathit{DEM}_{i,r,s,t}\mathit{DEM}_{i,r,s,t}}{P^M_{i,s,t}M_{i,s,t}}\\ \quad \mathit{SU}_{i,r,s,t}&=\frac{P^\mathit{DEM}_{i,r,s,t}\mathit{DEM}_{i,r,s,t}}{P^\mathit{DEMU}_{i,s,t}\mathit{DEMU}_{i,s,t}}\\ \quad \mathit{SV}_{i,r,s,t}&=\frac{P^\mathit{DEM}_{i,r,s,t}\mathit{DEM}_{i,r,s,t}}{P^\mathit{DEMV}_{i,s,t}\mathit{DEMV}_{i,s,t}}\\ \quad \mathit{Sh}_{i,r,s,t}&=\frac{P^\mathit{DEM}_{i,r,s,t}\mathit{DEM}_{i,r,s,t}}{P^\mathit{DEMTOT}_{i,s,t}\mathit{DEMTOT}_{i,s,t}} \end{array} $$

Mark-up

  • In domestic markets

$$ \quad \mathit{NB}_{i,r,t}(\mathit{EPD}_{i,r,t}+\frac{1}{\sigma_{\mathit{VAR}_i}})= \left[\frac{1}{\sigma_{\mathit{VAR}_i}}-\frac{1}{\sigma_{\mathit{ARM}_i}}\right] +\left[\frac{1}{\sigma_{\mathit{ARM}_i}}-\frac{1}{\sigma_{\mathit{GEO}_i}}\right]\mathit{SDU}_{i,r,t} +\left[\frac{1}{\sigma_{\mathit{GEO}_i}}-\frac{1}{\sigma_{\mathit{C}_i}}\right]\mathit{SDT}_{i,r,t} $$

  • In foreign markets (countries with the same level of development)

$$ \quad \mathit{NB}_{i,r,t}(\mathit{EP}_{i,r,s,t}+\frac{1}{\sigma_{\mathit{VAR}_i}}) =\left[\frac{1}{\sigma_{\mathit{VAR}_i}}-\frac{1}{\sigma_{\mathit{ARM}_i}}\right] +\left[\frac{1}{\sigma_{\mathit{IMP}_i}}-\frac{1}{\sigma_{\mathit{ARM}_i}}\right]\mathit{SE}_{i,r,s,t} +\left[\frac{1}{\sigma_{\mathit{ARM}_i}}-\frac{1}{\sigma_{\mathit{GEO}_i}}\right]\mathit{SU}_{i,r,s,t} +\left[\frac{1}{\sigma_{\mathit{GEO}_i}}-\frac{1}{\sigma_{\mathit{C}_i}}\right]\mathit{Sh}_{i,r,s,t} $$

  • In foreign markets (countries with different levels of development)

$$ \quad \mathit{NB}_{i,r,t}(\mathit{EP}_{i,r,s,t}+\frac{1}{\sigma_{\mathit{VAR}_i}}) =\left[\frac{1}{\sigma_{\mathit{VAR}_i}}-\frac{1}{\sigma_{\mathit{ARM}_i}}\right] +\left[\frac{1}{\sigma_{\mathit{IMP}_i}}-\frac{1}{\sigma_{\mathit{GEO}_i}}\right]\mathit{SV}_{i,r,s,t} +\left[\frac{1}{\sigma_{\mathit{GEO}_i}}-\frac{1}{\sigma_{\mathit{C}_i}}\right]\mathit{Sh}_{i,r,s,t} $$

Intervention price scheme (European Union)

  • Mode 0: no subsidy change

$$ \quad \mathit{TAXEXP}_{i,r,s,t}=\mathit{TAXEXP}_{i,r,s,{t_0}} $$

  • Mode 1: no subsidy

$$ \quad \mathit{TAXEXP}_{i,r,s,t}=0 $$

  • Mode 2:
    • Perfect competition $\quad P^Y_{i,r_\mathit{EU},t}=P^{\mathit{Interv}}_{i,r,t}$
    • Imperfect competition $\quad \sum_s \frac{P^Y_{i,r,t}}{1+\mathit{EP}_{i,r,s,t}}\mathit{TRADE}_{i,r,s,t}=P^{\mathit{Interv}}_{i,t}\sum_s \mathit{TRADE}_{i,r,s,t}$
  • Mode 3: subsidised exports ceiling

$$ \quad \sum_{s\neq r} \mathit{TRADE}_{i,r,s,t}= \mathit{MaxExpSub}_{i,r,t} $$

  • Mode 2 or 3, or subsidy change and subsidy for at least one destination before the change

$$ \quad \mathit{TAXEXP}_{i,r,s,t}=\mathit{TAXREF}_{i,r,t} \mathit{TAXEXP}_{i,r,s,{t_0}} $$

  • Mode 2 or 3, or subsidy change and no subsidy for all destinations before the change

$$ \quad \mathit{TAXEXP}_{i,r,s,t}=\mathit{TAXMOY}_{i,r,t} $$

  • Mode 2 or 3, or subsidy change

$$ \quad \mathit{TAXMOY}_{i,r,t}\sum_{s\neq r}\mathit{TRADE}_{i,r,s,t}=\sum_{s\neq r}\mathit{TAXEXP}_{i,r,s,t}\mathit{TRADE}_{i,r,s,t} $$

Comments:

The intervention price scheme in the EU is modelled as follows: as soon as the internal price becomes lower than the intervention price, the EU subsidises exports so as to raise the internal price to the level of the intervention price. In actual facts, the EU also increases inventories but inventories are not accounted for MIRAGE.

In practice, the price scheme is divided into 4 possible modes:

  • For countries other than the EU or sectors not concerned by intervention prices, the subsidy rate is exogenous.
  • When the intervention price is lower than the internal price, there is no export subsidy.
  • When the intervention price would be higher than the internal price, the export subsidy rate is endogenous. The distribution across importers is the same as in the baseline. If there was no subsidy in the baseline, this distribution is homogeneous.
  • The subsidization of exports is limited by a maximum of subsidized exports from the WTO. If this limit is reached, then this constraint replaces the price constraint.

When a simulation is complete, the model checks if the constraints defining a mode still hold. If they do not, then the mode is changed automatically until there is no more necessary change.

Investment

$$ \begin{array}{ll} \quad \mathit{INV}_{i,r,s,t}&=a_{i,r,s}B_{r,t}\mathit{KTOT}_{i,s,t}\:\mathrm{e}^{\alpha W^K_{i,s,t}}\\ \quad W^K_{i,r,t}&=P^K_{i,r,t}+P_{r,t} \mathit{TsubK}_{i,r,t}\\ \quad \mathit{INVTOT}_{s,t}&=\sum_{i,r}\mathit{INV}_{i,r,s,t} \end{array} $$

Regional equilibrium

$$ \quad \mathit{GDPVOL}_{r,t}*P^{\textit{CIndex}}_{r,t}=\mathit{REV}_{r,t}+\mathit{PIBMVAL}_t * \mathit{SOLD}_{r,t} $$

with $\quad P^{\textit{CIndex}}_{r,t} = {\prod\limits_i\left(\frac{P^C_{i,r,t}}{P^C_{i,r,{t_0}}}\right)^\frac{P^C_{i,r,{t_0}}C_{i,r,{t_0}}}{\sum\limits _j P^C_{j,r,{t_0}}C_{j,r,{t_0}}}}
$

$$ \begin{array}{ll} \quad \mathit{GDPVOL}_{r,t}*P^{\textit{CIndex}}_{r,t}=&\sum_s(\mathit{RQUOTA}_{r,s,t}-\mathit{RQUOTA}_{s,r,t})\\ &+\mathit{RECTAX}_{r,t}+\sum_i P^{\mathit{RN}}_{i,r,t}\mathit{RN}_{i,r,t}+\sum_{i,s}(P^K_{i,r,s,t}K_{i,r,s,t})\\ &+\overline{L}_{r,t}P^{\overline{L}_{r,t}}+\overline{\mathit{TE}}_{r,t} P^{\overline{\mathit{TE}}}_{r,t}+ \overline{H}_{r,t}P^{\overline{H}}_{r,t} \end{array} $$

$$ \begin{array} \quad \mathit{epa}_r\mathit{REV}_{r,t}=&\sum_{i,s}P^{\mathit{INVTOT}}_{s,t}\mathit{INV}_{i,r,s,t}\\ \quad \mathit{PIBMVAL}_{t}=&\sum_{i,r} P^{\mathit{VA}}_{i,r,t} \mathit{VA}_{i,r,t} \end{array} $$

Dynamics

$$ \begin{align} \quad K_{i,r,s,t}&=K_{i,r,s,t-1}(1-\delta)+\mathit{INV}_{i,r,s,t}\\ \quad \overline{\mathit{L}}_{r,t}&=\rho_r \overline{\mathit{L}}_{r,t-1}\\ \quad \overline{\mathit{H}}_{r,t}&=\rho_r \overline{\mathit{H}}_{r,t-1} \end{align} $$