MIRAGE (Online appendix)
This document presents the key elements of the MIRAGE1 model’s structure. The model’s equations are presented below. The documentation of the model consists of three papers: • Bchir, H., Decreux, Y., Gu´erin, J.-L., and Jean, S. (2002), ‘MIRAGE, a computable general equilibrium model for trade policy analysis’ CEPII Working Paper no 2002-17. • Decreux, Y., and Valin, H. (2007), ‘MIRAGE, an updated version of the model for trade policy analysis Focus on agriculture and dynamics’ CEPII Working Paper no 2007-15. • Fontagn´e, L., Four´e, J. and Ramos, M.-P. (2013) ‘MIRAGE-e: a general equilibrium long-term path of the world economy’ CEPII Working Paper no 2013-39. Supply Side On the supply side, each sector in MIRAGE is modeled as a representative firm, which combines value-added and intermediate consumption in fixed shares. Value-added is a CES bundle of imperfectly substitutable primary factors (capital, skilled and unskilled labor, land and natural resources). Firm’s demand for production factors is organized as a CES aggregation of land, natural resources, unskilled labor, and a bundle of the remaining factors. This bundle is a nested CES aggregate of skilled labor and capital (that are considered as relatively more complementary). MIRAGE assumes full employment of primary factors. Population, participation in the labor market and human capital evolve in each country (or region of the world economy) according to the demographics embedded in the macro projections. This determines the labor force as well as its skill composition (skilled/unskilled). Skilled and unskilled labor is perfectly mobile across sectors, but immobile between countries. Natural resources are sector specific, while land is mobile between agricultural sectors. Natural resources and total land for agricultural sectors are set at their 2007 levels: prices adjust demand to this fixed supply. Installed capital is assumed to be immobile (sector-specific), while investments are allocated across sectors according to their rates of return. The overall stock of capital evolves by combining capital formation and a constant depreciation rate of capital of 6% that is the same as in the long-term growth models. Gross investment is determined by the combination of saving (the saving rate from the growth model, applied to the national income) and the current account. Finally, while total investment is savingdriven, its allocation is determined by the rate of return on investment in the various activities. For simplicity, and because we lack reliable data on foreign direct investment at country of origin, host and sectoral levels, international capital flows only appear through the current account imbalances, and are not explicitly modeled. Demand side On the demand side, a representative consumer from each country/region maximizes instantaneous utility under a budget constraint and saves a part of its income, determined by saving rates projected in the long-term growth model. Expenditure is allocated to commodities and services according to a LES-CES (Linear Expenditure System – Constant Elasticity of Substitution) function. This implies that, above a minimum consumption of goods produced by each sector, consumption choices among goods produced by different sectors are made according to a CES function. This representation of preferences is well suited to our purpose as it is flexible enough to deal with countries at different levels of development. Within each sector, goods are differentiated by their origin. A nested CES function allows for a particular status for domestic products according to the usual Armington hypothesis (Armington, 1969): consumer’s and firm’s choices are biased towards domestic production, and therefore domestic and foreign 1 This
version is nicknamed MIRAGE-e 1.0 (1.0.1 – revision 97).
1
goods are imperfectly substitutable, using a CES specification. We use Armington elasticities provided by the GTAP database and estimated by Hertel et al. (2007). Total demand is built from final consumption, intermediate consumption and investment in capital goods. Dynamics Dynamics in MIRAGE are of two kinds: the total factor productivity is calibrated in a baseline exercise, while production factors dynamics are set exogenously. Both are built in MIRAGE using macroeconomic projections from the MaGE model documented in Four´e et al. (2013). Total factor productivity is based on the combination of three mechanisms. First, agri-food productivity is projected separately, as detailed in Fontagn´e et al. (2013). Second, a 2 percentage point growth difference between TFP in manufactures and services is assumed (as in van der Mensbrugghe, 2005). Third, the aggregate country-level TFP is calibrated in the baseline exercise in order to match both production factors and GDP projections resulting from the aggregate growth model, given the exogenous agri-food productivity and the productivity gap between manufacturing and services. Dynamics in MIRAGE is implemented in a sequentially recursive way. That is, the equilibrium can be solved successively for each period, given the exogenous trajectory for sector-specific TFP calibrated as described above, the accumulation of production factors – savings, current accounts, active population and skill level – coming from the growth model. Simulations extend up to 2025. Finally, MIRAGE is calibrated on the GTAP dataset version 8.1, with 2007 as a base year. References Armington, P.S. (1969), ‘A Theory of Demand for Products Distinguished by Place of Production’, Staff Papers - International Monetary Fund, 16(1): 159–178. Fontagn´e, L., Four´e, J. and Ramos, M.-P. (2013) ‘MIRAGE-e: a general equilibrium long-term path of the world economy’ CEPII Working Paper no 2013-39. Four´e, J., B´enassy-Qu´er´e, A., and Fontagn´e, L. (2013) ‘Modelling the world economy at the 2050 horizon’, Economics of Transition, 21(4): 617–654. Hertel, T.W., Hummels, D., Ivanic, M., and Keeney, R. (2007) ‘How confident can we be of CGEbased assessments of Free Trade Agreements?, Economic Modelling, 24(4): 611-635. Van der Mensbrugghe, D. (2005) LINKAGE Technical Reference Document, World Bank, Washington DC.
Options not included: • Energy in the value-added bundle. • Quality differentiation depending on the origin of goods. • Imperfect competition. • Carbon policy.
1
Notation
1.1
Variable names
Any variable X in MIRAGE will be associated with its price P X , unless explicited otherwise. In addition, we use several conventions: • EV oleX will denote the counterpart of variable X measured in energy quantity (Mtoe) • EmCO2X will denote the counterpart of variable X measured in quantity of CO2 emissions (M tCO2 ) ? • X ? will denote variable X measured at initial prices (ex. : GDPr,t and GDPr,t ).
2
1.2
Indexes
Regarding indexes, we will use the following notations: • i and j will refer to sectors. i will be used preferentially for goods while j will represent sectors. • r and s will represent regions.When appropriate r will denote the origin while s will represent the destination. r∗ will correspond to the reference region (the first one). • t will denote time (in years). The reference year is indexed by t0 .
1.3
Functional forms
Any relation between two variables A and B forming a bundle X will be parametrized by share or scale coefficients αA and αB . When appropriate, the elasticity of substitution between A and B inside X will be noted σX . In a nutshell, we will use the following abreviations for functional forms: • X ≡ Leontief [A; B] for Leontief-like relationships, • X ≡ CES σX [A; B] for Constant Elasticity of Substitution, • and X ≡ CD [A; B] for Cobb-Douglas. More than two components Our functional form will in many case have more than two com ponants. We then will add the other components in the notation, as in CD A, P A ; B, P B ; C,P C . However, if these components can be indexed by subscript k, then we will write X, P X ≡ CESkσX Ak , PkA , Leontiefk Ak , PkA or CDk Ak , PkA .
1.4
Booleans
We differentiate sectors by using booleans. For instance, if SET represents only some sectors, SET (i) will be true only for sectors in SET . (f alse otherwise) We can also write i ∈ SET or i ∈ / SET to denote inclusion of i in SET .
2
Parameters
Booleans T rT (i) Agri(i) Serv(i) Supply σiIC σiV A σiV AQL σiQ Demand cmini,r µj,r,s P W Oi σrC σ KG σiIM P σiARM
i is a transportation sector i is an agricultural sector i is a services sector
Elasticity Elasticity Elasticity GTAP) Elasticity
of substitution between intermediate consumption (σiIC = 0.6) of substitution between first-level value added (σiV A = 1.1) of substitution between second-level value added components (σiV AQL from of substitution between Skilled Labor and Capital (σiQ = 0.6)
Minimal consumption level for LES-CES (calibrated) Transport demand per unit of volume (calibrated) Normalisation parameter for world average price (calibrated) Elasticity of substitution between final consumptions (calibrated) Elasticity of substitution between capital goods (σ KG = 0.6) Elasticity of substitution between foreign origins (σiIM P from GTAP) Elasticity of substitution between domestic and foreign √ (σiARM = (σiIM P − 1)/ 2) + 1) 3
good
Factor markets TotalLandr0 Initial land supply (from GTAP) σ Land Land elasticity of transformation (σ Land = 0.5) δr Capital depreciation rate (δr = 0.06) Taxes and equivalents taxIC Tax on intermediate consumption i,j,r,t C taxi,r,t Tax on intermediate consumption KG taxi,r,t Tax on capital good consumption Tax on production taxP i,r,t Land subfi,r,t Subsidy to land use U nSkL subfi,r,t Subsidy to unskilled labor SkL subfi,r,t Subsidy to skilled labor Capital subfi,r,t Subsidy to capital T ariffi,r,s,t Import duty (and tariff-equivalent for NTMs when appropriate) tCosti,r,s,t Iceberg cost (for time spent in customs, NTMs, etc.) Export tax from GTAP (and export tax equivalent for NTMs when appropriate) taxEXP i,r,s,t AM F taxi,r,s,t Export tax equivalent to Multi-fiber agreement quotas (from GTAP) Energy and CO2 emissions εYe,r Conversion coefficient C Conversion coefficient εe,r εIC Conversion coefficient e,j,r D Conversion coefficient εe,r DEM εe,j,r Conversion coefficient κH Conversion coefficient e,r IC κe,j,r Conversion coefficient
for for for for for for for
energy content of production (Mtoe) energy content of final consumption (Mtoe) energy content of intermediate consumption (Mtoe) energy content of domestic demand (Mtoe) energy content of foreign demand (Mtoe) CO2 content of final consumption (MtCO2 ) CO2 content of intermediate consumption (MtCO2 )
Revenues and macroeconomic closure Savr,t Savings rate ai,r,s Investment initial scale coefficient α Elasticity of investment to return on capital (α = 40) Dynamics L gr,t H gr,t ∆Savingsr,t EP rod gr,t GDP gr,t T F P Agrij,r,t ∆gjT F P ∆CABalr,t
3
Exogenous growth rate of unskilled labor (from EconMap) Exogenous growth rate of skilled labor (from EconMap) Exogenous variation in savings rate (pct. of GDP, from EconMap) Exogenous growth rate of energy productivity (from EconMap) Exogenous growth rate of GDP (from EconMap) Exogenous growth rate of agricultural TFP Exogenous gap between industry and services productivity growth rate (∆gjT F P = 0.02 if j ∈ Serv, ∆gjT F P = 0 otherwise) Exogenous variation in current account balance (pct. of world GDP, from EconMap)
Variables
Supply
4
First Level Yi,r,t V Ai,r,t CN T ERi,r,t
Output of sector i Value added Aggregate intermediate consumption
Factors Landi,r,t N atResi,r,t RESVi,t U nSkLi,r,t SkLi,r,t Capitali,r,t
Land factor Natural resources Natural resources adjustment coefficient Unskilled labor Skilled labor Capital
Aggregates V AQLi,r,t Qi,r,t
Unskilled labor and Q aggregate Skilled labor and Capital aggregate
Intermediate consumption ICi,j,r,t Intermediate consumption of good i by sector j Demand Final demand Ur,t Ci,r,t BU DCr,t
Consumer utility Final consumption of good i Budget allocated to consumption
Capital good IN V T OTr,t IN Vi,r,s,t KGi,r,t
Total investment in region r Investment from r to sector i in s Capital demand for good i
Aggregate demand DEM T OTi,s,t Total demand for good i in region s Mi,s,t Demand for imported good i Di,s,t Domestic demand for good i DEMi,r,s,t Demand in region s for good i from region r T RADEi,r,s,t Exports of good i from region r to region s Transport T Ri,r,s,t Transport demand to route good i from r to s T RM odej,i,r,s,t Demand for transport type j to route good i from r to s T RSupplyj,r,t Supply of transport type j W orldT Rj,t Aggregate supply of transport mode j Prices F OB Pi,r,s,t CIF Pi,r,s,t W ORLD Pi,t
Free On Bord price Price including Cost, Insurance and Freight World average price for good i
5
Factor markets TotalU nSkLr,t Total supply of unskilled labor TotalSkLr,t Total supply of skilled labor TotalLandr,t Total supply for land TotalCapitalr,t Total capital supply Land wj,r,t Land return rate in sector j TotalLand wj,r,t Land return rate Capital Capital return rate in sector j wj,r,t TotalCapital wr,t Capital return rate U nSkL wr,t Wage for unskilled labor SkL wr,t Wage for skilled labor Energy and CO2 emissions Energy in Mtoe AgConse,r,t Quantity conservation adjustment coefficient (consumption side) AgDeme,r,t Quantity conservation adjustment coefficient (demand side) EV oleYe,r,t Production of energy e (Mtoe) EV oleCe,r,t Final consumption of energy e (Mtoe) EV oleICe,j,r,t Intermediate consumption of energy e (Mtoe) EV oleConse,r,t Total consumption of energy e in region r (Mtoe) EV oleDe,r,t Domestic demand for energy e (Mtoe) EV oleDEMe,r,s,t Foreign demand for energy e (Mtoe) EV oleDEM T OTe,s,t Total demand for energy e in region s EV oleDEM f romRege,r,tTotal demand for energy e from region r CO2 emissions EmCO2 ICe,j,r,t EmCO2 He,r,t
CO2 emissions from intermediate consumption of energy e in sector j (MtCO2 ) CO2 emissions from finale consumption of energy e (MtCO2 )
Revenues and macroeconomic closure Revenues P rodT axREVi,r,t ExpT axREVi,r,t T ariff REVi,s,t ConsT axREVi,s,r T axREVs,t REVr,t Closure Br,t CABalr,t
Revenue from production tax Revenue from export tax Revenue from tariffs Revenue from consumption tax Total tax revenues Total revenues
Investment scale coefficient Current account balance
6
GDP and numeraire NA Domestic demand (National Accounts method) Di,r,t NA DEMi,r,s,t Foreign demand (National Accounts method) NA DEM T OTi,s,t Total demand (National Accounts method) N A? DEM T OTi,s,t Total demand at initial prices (National Accounts method) A KGN Capital good demand (National Accounts method) i,r,t NA Ci,r,t Final consumption (National Accounts method) T F P Jj,r,t Sector-specific component of TFP T F Pr,t National component of TFP GDPr,t Gross Domestic Product ? GDPi,r,t Gross Domestic Product at initial prices W GDP V ALt World GDP
4 4.1
Equations Supply
First-stage in production function Yi,r,t ≡ Leontief [V Ai,r,t ; CN T ERi,r,t ]
(1)
Intermediate consumption σ IC
CN T ERi,r,t ≡ CESj i [ICi,j,r,t ] IC DEM T OT Pi,j,r,t = Pi,r,t 1 + taxIC i,j,r,t
Value added V Ai,r,t
VA
(T F Pr,t T F P Jj,r,t )
σiV A
≡ CES σi
V AQL
V AQLi,r,t ≡ CES σi Qi,r,t ≡ CES
σiQ
(3)
[Landi,r,t ; N atResi,r,t RESVi,t ; V AQLi,r,t ]
[U nSkLi,r,t ; Qi,r,t ]
(4) (5)
[SkLi,r,t ; Capitali,r,t ]
EIC DEM T OT Pi,j,r,t = Pi,r,t 1 + taxIC i,j,r,t
4.2
(2)
(6)
(7)
Demand
Final demand C
Ur,t ≡ CESiσ [Ci,r,t − cmini,r ] X C BU DCr,t = Pi,r,t Ci,r,t
(8) (9)
i C DEM T OT Pi,r,t = Pi,r,t 1 + taxC i,r,t
(10)
Capital good IN V T OTs,t ≡ CESiσ KG DEM T OT Pi,r,t = Pi,r,t
KG
[KGi,s,t ] 1 + taxKG i,r,t
(11) (12)
Aggregate demand DEM T OTi,s,t = Ci,s,t +
X
ICi,j,s,t
(13)
j
7
DEM T OTi,r,s ≡ CES σ σiIM P
Mi,r,s ≡ CESr
ARM
[Di,s,t ; Mi,s,t ]
(14)
[DEMi,r,s,t ]
(15)
Transport T RADEi,r,s,t = DEMi,r,s,t
(16)
T Rj,r,s,t = µj,r,s (1 + tCosti,r,s,t ) T RADEj,r,s,t
(17)
T Rj,r,s,t ≡ CDi∈T rT [T RM odei,j,r,s,t ]
(18)
WorldT Ri,t ≡ CDr [T RSupplyi,r,t ] if i ∈ T rT X WorldT Ri,t = T RM odei,j,r,s if i ∈ T rT
(19) (20)
j,r,s
Prices F OB Y Pi,r,s,t = Pi,r,t (1 + tCosti,r,s,t ) 1 + taxP i,r,t
AM F 1 + taxEXP i,r,s,t + taxi,r,s,t
(21)
CIF F OB TR Pi,r,s,t = Pi,r,s,t + µi,r,s Pi,r,s,t (1 + tCosti,r,s,t ) D Y Pi,r,t = Pi,r,t 1 + taxP i,r,t
(22)
DEM Pi,r,s,t
(24)
=
W ORLD Pi,t
CIF Pi,r,s,t
(1 + T ariffi,r,s,t )
1 " # P T RADE i,r,s,t r,s Y 1 CIF T RADEi,r,s,t Pi,r,s,t = P W Oi r,s
Commodity market equilibrium ( P Di,r,t + s DEMi,r,s,t Yi,r,t = P Di,r,t + s DEMi,r,s,t + T RMi,r,t
4.3
(23)
if i ∈ / T rT if i ∈ T rT
(25)
(26)
Factor markets
Labor TotalU nSkLr,t =
X
U nSkLj,r,t
(27)
j
TotalSkLr,t =
X
SkLj,r,t
(28)
j
Land TotalLandr,t ≡ CETiσ TotalLandr,t TotalLandr,t
Land
Land wi,r,t , Landi,r,t !σTotalLand TotalLand w r,t = T otalLandr0 U Pr,t X = Landj,r,t
(29) (30) (31)
j Land TotalLand wj,r,t = wr,t
(32)
Capital Stock and investment Ki,r,s,t = Ki,r,s,t−1 (1 − δr ) + IN Vi,r,s,t X Capitali,s,t = Ki,r,s,t
(33) (34)
r
TotalCapitalr,t =
X
Capitalj,r,t
(35)
j
8
Factor-based subsidies Land Land U Land Pi,r,t = wi,r,t − Pr,t subfi,r,t
(36)
U nSkL U nSkL U U nSkL Pi,r,t = wr,t − Pr,t subfi,r,t
(37)
SkL SkL U SkL Pi,r,t = wr,t − Pr,t subfi,r,t
(38)
Capital Pi,r,t
4.4
=
Capital wi,r,t
Capital U Pr,t subfi,r,t
−
(39)
Energy and CO2 emissions
Energy in Mtoe Production EVoleYe,r,t = εYe,r . Ye,r,t
(40)
Consumption EVoleCe,r,t = AgConse,r,t . εC e,r . Ce,r,t EVoleIC e,j,r,t = AgConse,r,t .
εIC e,j,r
EVoleConse,r,t = EVoleCe,r,t +
X
(41)
. ICe,j,r,t
(42)
EVoleICe,j,r,t
(43)
j
Demand EVoleDe,r,t = AgDeme,r,t . εD e,r . De,r,t
(44)
EVoleDEMe,r,s,t = AgDeme,r,t . εDEM e,r,s . DEMe,r,s,t X EVoleDEM T OTe,s,t = EVoleDe,s,t + EVoleDEMe,r,s,t
(45) (46)
r
EVoleDEMf romRege,r,t = EVoleDe,r,t +
X
EVoleDEMe,r,s,t
(47)
s
Quantity accounting EVoleYe,r,t = EVoleDEMf romRege,r,t
(48)
EVoleConse,r,t = EVoleDEM T OTe,r,t
(49)
CO2 emissions
4.5
EmCO2 ICe,j,r,t = AgConse,r,t . κIC e,j,r . ICe,j,r,t
(50)
EmCO2 He,r,t = AgConse,r,t . κH e,r . Ce,r,t
(51)
Revenues and macroeconomic closure
Revenues Production tax Y P rodT axREVi,r,t = taxP i,r,t . Pi,r,t . Yi,r,t
(52)
Export tax ExpT axREVi,r,t =
X
AM F P Y taxEXP i,r,s,t + taxi,r,s,t . 1 + taxi,r,t (1 + tCosti,r,s,t ) Pi,r,t . T RADEi,r,s,t
s
(53) Tariff T ariff REVi,s,t =
X
CIF T ariffi,r,s,t . Pi,r,s,t . T RADEi,r,s,t
r
9
(54)
Consumption tax DEM T OT ConsT axREVi,s,t = taxC . Ci,s,t i,s,t . Pi,s,t DEM T OT + taxKGC . KGi,s,t i,s,t . Pi,s,t X DEM T OT + taxIC . ICi,j,s,t i,j,s,t . Pi,s,t
(55)
j
Total revenue X
T axREVs,t =
P rodT axREVi,s,t + ExpT axREVi,s,t + T ariff REVi,s,t + ConsT axREVi,s,t
i
(56) REVr,t =
X
N atRes Land SkL [Pi,r,t N atResi,r,t + Pi,s,t Landi,s,t + Pi,s,t SkLi,s,t
i U nSkL + Pi,s,t U nSkLi,s,t +
X
P Capitali,s,t Ki,r,s,t ]
s
+ T axREVr,t
(57)
BU DCr,t = (1 − Savr,t ) REVr,t
(58)
Closure " IN Vi,r,s,t = Br,t ai,r,s Capitali,s,t exp α X
IN V T OTs,t =
Capital wi,s,t − δr IN V T OT Ps,r
!# (59)
IN Vi,r,s,t
(60)
i,r
X
Savr,t REVr,t =
IN V T OT Ps,t IN Vi,r,s,t + W GDP VALt . CABalr,t
(61)
i,s
GDP and numeraire National Accounts NA D NA D Pi,r,t , Di,r,t = Pi,r,t , Di,r,t DEM Pi,r,s,t
NA
(62)
DEM = Pi,r,s,t
(63) NA
NA D NA DEM T OTi,s,t = Pi,s,t Di,s,t +
X
DEM Pi,r,s,t
NA
T RADEi,r,s,t
(64)
r DEM T OT Pi,s,t
NA
=
NA DEM T OTi,s,t N A? DEM T OTi,s,t
KGN A A Pi,r,t , KGN i,r,t
CNA NA Pi,r,t , Ci,r,t
GDPr,t =
X
=
=
(65)
DEM T OT N A Pi,s,t
DEM T OT N A Pi,s,t
NA
C NA KG Pi,r,t Ci,r,t + Pi,r,t
1+
NA
1+
taxKG i,r,t
taxC i,r,t
,
!
KG Pi,r,t KG Pi,r,t
NA
C Pi,r,t , C N A Ci,r,t Pi,r,t
KGi,r,t
(66)
! (67)
A KGN i,r,t
i
+
X
Y Pi,r,t 1 + taxP i,r,t T rSupplyi,r,t
i∈T rT (i)
+
X
F OB CIF Pi,r,s,t T RADEi,r,s,t − Pi,s,r,t T RADEi,s,r,t
i,s
10
(68)
Numeraire W GDP VALt =
X
GDPr,t
(69)
r
X
? GDPr,t = W GDP V ALt
(70)
r
4.6
Dynamics
Exogenous variables L TotalU nSkLr,t = 1 + gr,t TotalU nSkLr,t−1 H TotalSkLr,t = 1 + gr,t TotalSkLr,t−1
(71)
Savr,t = Savr,t−1 + ∆Savingsr,t
(73)
(72)
Baseline ? GDP ? GDPr,t,ref = 1 + gr,t GDPr,t−1,ref ( T F P Agrij,r,t T F P Jj,r,t,ref . T F Pr,t,ref = 1 + ∆gjT F P T F P Jj,r,t−1,ref . T F Pr,t,ref CABalr,t,ref = CABalr,t−1,ref + ∆CABalr,t
(74) if j ∈ Agri if j ∈ / Agri
(75) (76)
Simulation T F P Jj,r,t,sim = T F P Jj,r,t,ref
(77)
T F Pr,t,sim = T F Pr,t,ref
(78)
CABalr,t,sim = CABalr,t,ref
(79)
11
