A SEQUENTIAL DYNAMIC CGE MODEL FOR
POVERTY ANALYSIS
Nabil Annabi*, John Cockburn* and Bernard Decaluwé*
Preliminary Draft1 (May, 2004)
* CIRPEE, Laval University, http://www.cirpee.org
1
This draft was prepared for the Advanced MPIA Training Workshop in Dakar, Senegal, June 10-14,
2004. For more information see the PEP Network website http://www.pep-net.org
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
1. Introduction
Much current debate focuses on the role of growth in alleviating poverty. However, the
majority of CGE models used in poverty and inequality analysis are static in nature. The inability of
this kind of model to account for growth effects makes them inadequate for medium run analysis of
the poverty impacts of economic policies. They exclude accumulation effects and do not allow the
study of the transition path of the economy where short run policy impacts are likely to be
different from those of the medium run2. To overcome this limitation we propose a simple dynamic
CGE model for poverty analysis.
Dynamic general equilibrium models can be classified as truly dynamic (“intertemporal”) or
sequential dynamic (“recursive”) models. Truly dynamic models are based on optimal growth
theory where the behaviour of economic agents is characterized by perfect foresight. They know
all about the future and react to future changes in prices. Households maximize their
intertemporal utility function under a wealth constraint to determine their consumption schedule
over time. Investment decisions by firms are the result of cash flow maximization over the whole
time horizon. However, the application of this kind of model to the analysis of poverty and
inequality is not straightforward and remains in our research agenda for the future. In this
document, we develop a sequential dynamic CGE model for poverty analysis. This kind of dynamics
is not the result of intertemporal optimization by economic agents. Instead, these agents have
myopic behaviour. A sequential dynamic model is basically a series of static CGE models that are
linked between periods by an exogenous and endogenous variable updating procedure. Capital
stock is updated endogenously with a capital accumulation equation and population (total labour
supply) is updated exogenously between periods. It is also possible to add updating mechanisms for
other variables such as public expenditure, transfers, technological change or debt accumulation.
But the crucial questions to deal with in this kind of model concern the distribution of new
investments between sectors and the capital stock calibration. These will be discussed below.
This document is organized as follows. We begin by a short overview of the static version of the
model in section 2 and present its extension to include sequential dynamics in section 3. Sections 4
and 5 describe briefly the calibration and the resolution of the model. Finally, we present a
complete list of the equations and the GAMS codes in Appendix I and II.
2
Annabi (2003) uses an intertemporal model to study the effects of trade liberalization in Tunisia. Simulation results
show that the static version of the model underestimates trade liberalization’s impacts on production and welfare,
since it excludes accumulation effects.
2
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
2. Static module
The static part of the model is based on the EXTER+ model3. As the main objective of the
present document is to introduce sequential dynamics we present only a brief description of
production, consumption, international trade and government behaviour.
2.1 Production
We adopt a multi-stage production function. In a first stage, sectoral output is a Leontief
function of value added and total intermediate consumption. Value added is in turn represented by
a CES function of labour and capital in the non-agricultural sectors (industry and services), and a
CES function of land and a composite factor in agriculture. The latter is also represented by a CES
function of primary factors: agricultural capital and labour. Value added in the public sector is
generated by labour alone. In the different production activities we assume that a representative
firm remunerates factors of production and pays dividends to households.
2.2 Consumption
Household demand is represented by a linear expenditure system (LES) derived from the
maximization of a Stone–Geary utility function. The model includes four household categories: rural
poor, urban poor, rural rich and urban rich. Minimal consumption levels are calibrated using
estimates of the income elasticity and the Frisch parameters.
2.3 Foreign Trade
We assume that foreign and domestic goods are imperfect substitutes. This geographical
differentiation is introduced by the standard Armington assumption with a constant elasticity of
substitution function (CES) between imports and domestic goods. On the supply side, producers
make an optimal distribution of their production between exports and domestic sales according to
a constant elasticity of transformation (CET) relation.
2.4 Government
The government receives direct tax revenue from households and firms and indirect tax
revenue on domestic, imported and exported goods. Its expenditure is allocated between the
consumption of goods and services, public wages and transfers. The model accounts for indirect or
direct tax compensation in the case of a tariff cut.
3
For a general presentation of EXTER+ model see Cockburn et al., (2004).
3
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
2.5 Equilibrium
General equilibrium is defined by the equality (in each period) between supply and demand of
goods and factors and the investment-saving identity. In a future version, the model will be
extended to take into account unemployment and endogenous labour supply. Finally, the nominal
exchange rate is the numéraire.
3. Dynamic module: EXTER+SD model
The EXTER+SD (Sequential Dynamic) model is formulated as a static model, which is solved
sequentially over time. In every period the capital stock is updated with a capital accumulation
equation. Total labour supply increases at the same rate as exogenous population growth. We also
account for debt accumulation. We use the static model, index all variables in time (t) and
introduce the equations presented in Table 14:
Table 1: Additional equations for the sequential dynamic model
1.
2.
KDit +1 = (1 − δ ) KDit + Indit
Capital Accumulation
LSt +1 = ( 1 + ng ) ⋅ LSt
Labour force growth
2
3.
4.
⎛r ⎞
⎛r ⎞
Ind it
= γ 1i ⎜ it ⎟ + γ 2i ⎜ it ⎟
KDit
⎝ Ut ⎠
⎝ Ut ⎠
U t = Pinvt ⋅ ( ir + δ )
min
tr , h , t +1
Investment demand
Capital user cost
= ( 1 + ng ) ⋅ C
5.
C
6.
ITt = Pinvt ⋅ ∑ Indit
min
tr , h ,t
LES minimal consumption
Investment equilibrium
i
where,
KDit
Indit
LSt
SGt
CABt
rit
PCit
: Capital stock
: Investment by destination
: Total labour supply
: Government savings
: Foreign savings
: Return to capital
: Consumption price
DGt
DFt
δ
ng
ir
: Public debt
µi
ITt
: Share parameter (free parameter)
: Foreign debt
: Capital depreciation rate (free parameter)
: Population growth rate (free parameter)
: Interest rate (free parameter)
: Nominal total investment
3.1 Capital accumulation
Equation 1 describes the law of motion for the sectoral capital stock. It supposes implicitly that
the stocks are measured at the beginning of the period and that the flows are measured at the end
4
See Appendix I for the complete list of equations.
4
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
of the period. We note that in some cases the law of motion can be applied to the total capital
stock KSt , rather than at the sectoral level.
3.2 Growth in labour supply
With the introduction of equation 2, total labour supply LSt becomes an endogenous variable,
although it simply increases at the exogenous rate ng , which is simultaneously the population
growth rate and the labour force growth rate. Note that the minimal level of consumption within
the LES function also increases at the same rate ng (equation 5).
3.3 Investment demand
As we mentioned in the introduction the question is how will new investment be distributed
between different sectors? This can be done either through a distribution function or an
investment demand function. Note that the investment here is not the investment by origin
(product) that we have already in the static model, but it is investment by sector of destination.
The investment demand function we use here is similar to the one proposed by Bourguignon,
Branson and De Melo (1989). In equation 3,
γ1i
and
γ 2i
are positive parameters calibrated on the
basis of the investment elasticity and the investment equilibrium equation (equation 8). The
investment rate is increasing with respect to the ratio of the rate of return to capital and its user
cost.
In introducing investment by destination, we respect the equality condition with total
investment by origin in the original social accounting matrix (equation 6). Given the investment
demand equation,
2
⎛r ⎞
⎛r ⎞
Ind it
= γ 1i ⎜ it ⎟ + γ 2i ⎜ it ⎟
KDit
⎝ Ut ⎠
⎝ Ut ⎠
the elasticity of the investment rate with respect to the ratio of the rate of return of capital to its
user cost is:
d ⎛⎜
⎝
⎞
⎛ rit ⎞
⎜ U ⎟
KDit ⎟⎠
t⎠
⋅ ⎝
=ν
rit ⎞
Indit
⎛
⎛
⎞
d⎜
⎟
⎜
KDit ⎟⎠
⎝ Ut ⎠
⎝
Indit
which implies that:
5
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
γ 2i
2 −ν
=
ν − 1 γ ⎛ rit ⎞
1i ⎜ U ⎟
t⎠
⎝
The parameters
γ1i and γ 2i
are therefore positive only if the value of the elasticity of
Indit
with
KDit
rit
is between 1 and 2. If ν = 1.5 we obtain the following relation:
Ut
respect
γ 2i = γ 1i
rit
Ut
this relation, taken together with the investment demand equation, makes it possible to calibrate
γ 1i and γ 2i .
In the literature on dynamic CGE models, we note some alternative assumptions for the investment
demand function or the distribution of total investment. Abbink, Braber and Cohen (1995), present
an applied sequential dynamic CGE model for Indonesia. They determine the shares of investment
by destination as follows:
θi 0 ⎛⎜
PRit ⎞
⎟
⎝ APRt ⎠
θit =
PR
∑θi 0 ⋅ ⎛⎜⎝ APRitt ⎞⎟⎠
i
where
θi 0 , PRit
and APRt are respectively the base run sectoral investment shares (which are
assumed equal to the base year sectoral shares in total capital remuneration), sectoral profit rates
and the average profit rate defined as follows:
θi 0 =
ri 0 KDi 0
∑ ri 0 KDi 0
i
rit ⋅ KDit − δ ⋅ KDit ⋅ Pinvt
KDit ⋅ Pinvt
KDit ⋅ Pinvt
APRt = ∑
⋅ PRit
i ∑ j KD jt ⋅ Pinvt
PRit =
Finally, capital accumulation is written as:
KDit +1 = (1 − δ ) KDit + θit ⋅ ITt
6
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
ITt
where
is the total volume of investment. The second element on the right hand side of this
expression represents investment by sector of destination.
Jung and Thorbecke (2003) use the following investment demand function:
⎛
⎞
Indit
KINCit
= Ai ⋅ ⎜
⎟
KDit
⎝ Pinvt ⋅ KDit ⋅ ir ⎠
βi
with KINCit = rit KDit this equation could be simplified to:
⎛
⎞
Indit
rit
= Ai ⋅ ⎜
⎟
KDit
⎝ Pinvt ⋅ ir ⎠
Where
Ai ,r
and
βi
βi
represent a scale parameter, the interest rate and the investment elasticity,
the latter of which is fixed to 1.
3.4 Alternative version with debt accumulation
An alternative or extended version of this sequential dynamic model can be obtained by adding
the following equations for public and foreign debt:
DGt +1 − DGt = irDGt − SGt
DFt +1 − DFt = irDFt + CABt
The variation in public debt is equal to the interest payments on this debt less public savings. In
the same manner, the variation in foreign debt is equal to interest payments on this debt plus the
current account balance (foreign savings):
In this version, we assume that firms earn the interest paid on bonds issued by the government
to finance its deficit. The government is the only agent who can borrow abroad and therefore they
pay the interest on foreign debt, as well as on their public debt. In this second case, we obtain:
Firm income:
(
YFt = 1 − λ − λ ROW
Public savings:
) ∑ r KD
it
it
+ irDGt
i
SGt = YGt − Gt − TGt − irDGt − irDFi
7
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
The current account balance:
CABt = et ∑ PWM i M it + λ ROW ∑ rit KDit + DIV ROW − et ∑ PWEi EX it + irDFt
i
i
i
If initial debt is nil, the debt equations may be rewritten as:
DGt +1 = DGt − SGt
DFt +1 = DFt + CABt
Debts are exogenous in the first period. Labour supply and capital demand are endogenous, except
in the first period.
It is interesting to note that, with the introduction of the debt equations, we can change the
traditional closure rules of static CGE models. For example, we can fix the ratio of public or foreign
debt with respect to GDP, instead of fixing the public or foreign deficits.
4. Calibration
With the introduction of the investment demand and the accumulation equations we must
adjust the calibration of the capital stock to make it consistent. For this purpose we make
assumptions on the sectoral growth rates of capital, as data is not generally available on
investment by destination in developing countries. Some possible assumptions include:
•
•
The investment rate is equal to the production growth rate.
The investment rate is equal to the sum of the population growth rate and the
depreciation rate5.
In our application we fixed the sectoral growth rates of capital at 5 percent in order to
calibrate the capital stock6. We call this “method 1”. The second possible way of calibrating the
capital stock (method 2) consists of making an assumption about the rate of return to capital and
calibrating its stock from the operating surplus (OS) in the social accounting matrix. Table 3
presents the advantages and disadvantages of each method.
5
The latter assumption implies the steady state condition for capital accumulation, Indit Kdit = ( ng + δ ) that is used
in intertemporal models.
Note that in the sequential dynamic models it is not possible to normalize the return to capital otherwise one could
obtain unrealistic growth trends.
6
8
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
Table 3: The capital stock calibration methods
Advantages
Method 1
(assumed
sectoral
growth
rates)
Method 2
(assumed
sectoral
rates of
return)
Disadvantages
- Accumulation profile more
homogenous.
- If OSi are not well estimated, ri could differ
substantially between sectors.
- If OSi < 0 fl ri < 0: acceptable.
- If Indi < 0 fl Ki < 0: not acceptable.
- Base run Ri are equal (long run
equilibrium).
- If the OSi are not well estimated, Ki (and Indi/Ki)
could differ substantially between sectors.
- If Indi < 0 fl Ki > 0.
- If the OSi < 0 fl Ki < 0: not acceptable.
5. Model Execution
The model is solved simultaneously as a system of non linear equations. To illustrate, we
simulate the impact of a total tariff cut in the industrial sector and present some results7. The
values of the new (free) parameters of the model are presented in the following table:
Table 2: Free parameters
δ
ir
0.025
ν
ng
1.50
0.05
0.03
In static CGE models, counterfactual analysis is made with respect to the base run that is
represented by the initial SAM. But in dynamic models the economy grows even without a policy
shock and the analysis should be done with respect to the growth path in the absence of any shock.
Note, in graph 1, that the economy grows even without a tariff cut. Graph 2 shows the evolution of
the ratio of the industrial capital stock in the presence of the policy shock with respect to capital
stock without the shock (“business as usual” or Bau).
7
This section will be developed in the future version of this document.
9
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
- Graph 1: Industrial capital stock with and without shock 45
Base run = 1
40
35
30
25
20
1
2
3
4
5
6
7
8
9
10
Years
Shock
Bau
- Graph 2: Ratio of industrial capital stocks with and without shock 1.1
1.09
1.08
Base run = 1
1.07
1.06
1.05
1.04
1.03
1.02
1.01
1
1
2
3
4
5
6
7
8
9
10
Years
IND
10
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
References
Annabi N. (2003), Libéralisation Commerciale en Tunisie: Une analyse à l’Aide de Modèle d’Équilibre
Général Calculable Dynamique. Ph.D. dissertation, University of Paris I (Sorbonne).
Abbink G. A., Braber M. C. et Cohen S. I. (1995), A SAM-CGE demonstration model for Indonesia: A
Static and Dynamic specifications and expirements. International Economic Journal, Vol 9 (3),
p.15-33.
Agénor, P-R. (2003), The Mini-Integrated Macroeconomic Model for Poverty Analysis. The World Bank.
Ballard, C. L., Fullerton, D., Shoven, J. B. and Whalley, J. (1985), A General Equilibrium Model for Tax
Policy Evaluation. The University of Chicago Press.
Beghin, J., Dessus, S., Roland, H. D. and
Van der Mensbrugghe, D. (1996), General Equilibrium
Modelling of Trade and The Environment. OECD, Technical Paper No. 116.
Bourguignon, F., Branson, W. H. and de Melo, J. (1989), Macroeconomic Adjustment and Income
Distribution: A Macro-Micro Simulation Model. OECD, Technical Paper No.1.
Cockburn, J., Decaluwé, B. and Robichaud, V. (2004), Trade Liberalization and Poverty: A CGE Analysis
of the 1990s Experience, (forthcoming).
Decaluwé, B., Martens, A. and Savard, L. (2001), La Politique économique du développement et les
modèles d’équilibre général calculable. Les Presses de l’Université de Montréal.
Dervis, K., de Melo, J. and Robinson S. (1982), General Equilibrium Models for Development Policy,
Cambridge University Press.
Jung, H.S. and Thorbecke, E.(2003) The Impact of Public Education Expenditure on Human Capital,
Growth, and Poverty in Tanzania and Zambia: A General Equilibrium Approach. Journal of Policy
Modeling, 25: 701–725
Van der Mensbrugghe, D. (2003), LINKAGE. Technical Reference Document, The World Bank.
Schubert, K. (1993), Les modèles d’équilibre général calculable : une revue de la littérature. Revue
d’Économie Politique, 103 (6) : 775-825.
11
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
Appendix I: Equations of EXTER+SD
Model structure
The subscript presenting the time periods is introduced only when necessary.
Production
1.
⎡ CI j VA j ⎤
XS j = min ⎢
,
⎥
⎣ io j v j ⎦
4
(
)
−ρnag
−ρnag ⎤
KL ⎡ KL
KL
α nag LDnag
+ 1 − α nag
KDnag
2. VAnag = Anag
KL
⎣
3. VAAGR = AtrCL ⎡ αCL CF
−ρCL
⎣
4.
KL
⎦
+ ( 1 − αCL ) LAND
−ρCL
−ρagr ⎤
KL ⎡ KL
KL
α LD −ρagr + ( 1 − α agr
CF = Aagr
) KDagr
⎣ agr agr
⎦
KL
−1
KL
−1
⎤
⎦
KL
ρnag
−1
ρCL
KL
ρagr
2
1
1
5. VAntr = LDntr
1
6.
CI j = io j XS j
4
7.
DI tr , j = aijtr , j CI j
8.
⎛ 1 − αCL ⎞
LAND = ⎜
CL ⎟
⎝ α
⎠
9.
⎛ αtrKL ⎞
LDtr = ⎜
KL ⎟
⎝ 1 − αtr ⎠
10. LDNTR =
12
σCL
σtrKL
⎛ rc ⎞
⎜ rl ⎟
⎝ ⎠
⎛ rtr ⎞
⎜ w⎟
⎝ ⎠
σCL
CF
1
KDtr
3
σtrKL
PNTR XS NTR − ∑ PDtr DI tr ,NTR
tr
w
1
Income and savings
R
L
YH h = λW
h ⋅ w∑ LD j + λ h ∑ rtr KDtr + λ h ⋅ rl ⋅ LAND
j
11.
tr
4
+ PINDEX ⋅ TGh + DIVh
12. YDH h = YH h − DTH h
4
13. SH h = ν ⋅ ψ h ⋅ YDH h
4
14. YF = λ RF
∑ rtr KDtr + λ LF ⋅ rl ⋅ LAND
1
tr
12
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
15. SF = YF −
∑ DIVh
− e ⋅ DIV ROW − DTF
1
h
16. YG =
∑ TItr + ∑ TIEtr + ∑ TIM tr + ∑ DTH h + DTF
1
∑ TGh
1
tr
tr
17. SG = YG − G − PINDEX
tr
h
h
18. TI tr = txtr ( Ptr XStr − PEtr EX tr ) + txtr ( 1 + tmtr
) e PWM tr M tr
3
19. TIM tr = tmtr e PWM tr M tr
3
20. TIEtr = tetr PEtr EX tr
3
21. DTH h = tyhhYH h
4
22. DTF = tyf ⋅ YF
1
Demand
23. CTH h = YDH h − SH h
4
⎛
24. PC tr C tr , h = PC tr C trMIN
, h + γ tr , h ⎜ CTH h −
⎝
∑ PC
trj
trj
⎞
C trjMIN
,h ⎟
⎠
12
25. G = XS ntr Pntr
1
26. INVtr =
µtr IT
PCtr
3
27. DITtr =
∑ DI
3
j
j
Prices
Pj XS j − ∑ PCtr DI tr , j
28. PV j =
29. rnag =
tr
VAj
PVnagVAnag − w LDnag
KDnag
rc ⋅ CF − w LDAGR
KDAGR
PVAGRVAAGR − rl ⋅ LAND
31. rc =
CF
30. rAGR =
4
2
1
1
32. PDtr = ( 1 + txtr
) PLtr
3
33. PM tr = ( 1 + txtr
) ( 1 + tmtr ) e ⋅ PWM tr
3
e ⋅ PWEtr
1 + tetr
3
35. PCtr Qtr = PDtr Dtr + PM tr M tr
3
36. Ptr XStr = PLtr Dtr + PEtr EX tr
3
34. PEtr =
13
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
∏ ⎛⎜⎝
37. Pinv =
PCtr ⎞
µtr ⎟⎠
tr
µtr
1
∑ δ PV
38. PINDEX =
i
1
i
i
International Trade
39. XStr = BtrE ⎡⎣ βtrE EX trκtr + ( 1 − βtrE
E
⎡ ⎛ PEtr ⎞ ⎛ 1 − βtrE
40. EX tr = ⎢ ⎜
⎟⎜ E
⎣ ⎝ PLtr ⎠ ⎝ βtr
)
1
E
Dtrκtr ⎤⎦ κtr
E
3
τtrE
⎞⎤
⎟ Dtr
⎠ ⎦⎥
3
−1
41. Qtr = AtrM ⎡⎣ α trM M tr−ρtr + ( 1 − α trM ) Dtr−ρtr ⎤⎦ ρtrM
M
M
⎡ ⎛ PDtr ⎞ ⎛ α trM ⎞ ⎤
42. M tr = ⎢ ⎜
⎟⎜
M ⎟
⎣ ⎝ PM tr ⎠ ⎝ 1 − α tr ⎠ ⎦⎥
3
σtrM
3
Dtr
CAB = ∑ PWM tr M tr + λ ROW ∑ rtr KDtr e + λ LROW rl ⋅ LAND e
tr
43.
tr
1
+ DIV ROW − ∑ PWEtr EX tr
tr
Equilibrium
44. Qtr = DITtr +
∑C
tr ,h
+ INVtr
3
h
45. LS =
∑ LD
1
j
j
46. IT =
∑ SH
h
+ SF + SG + e ⋅ CAB
1
h
⎛⎛
⎞ ⎡ PCOtr ⎤
EVh = ⎜ ⎜ CTH h − ∑ PCtrj CtrjMIN
,h ⎟ ∏ ⎢
trj
⎝⎝
⎠ tr ⎣ PCtr ⎥⎦
47.
⎛
⎞⎞
− ⎜ CTHOh − ∑ PCOtrj CtrjMIN
,h ⎟ ⎟
trj
⎝
⎠⎠
γ tr ,h
4
Dynamic Equations
48.
KDtr ,t +1 = (1 − δ ) KDtr ,t + Indtr ,t
49. LSt +1 = ( 1 + ng ) ⋅ LSt
min
50. Ctrmin
, h , t +1 = ( 1 + ng ) ⋅ Ctr , h ,t
3
1
12
14
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
2
⎛R ⎞
⎛R ⎞
51.
= γ 1tr ⎜ tr ,t ⎟ + γ 2tr ⎜ tr ,t ⎟
KDtr ,t
⎝ Ut ⎠
⎝ Ut ⎠
52. U t = Pinvt ⋅ ( ir + δ )
Ind tr ,t
53. ITt = Pinvt ⋅
3
1
∑ Indtr ,t
1
tr
Total: 151
Endogenous variables
Number of variables
Ctr ,h :
Household h's consumption of good tr (volume)
CF :
CI j :
Composite agricultural capital-labor factor (volume)
1
Total intermediate consumption of activity j (volume)
4
Household h's total consumption (value)
4
Demand for domestic good tr (volume)
3
CTH h :
Dtr :
DI tr , j :
DITtr :
DTF :
DTH h :
e :
EVh :
EX tr :
G :
INVtr :
IT :
LD j :
M tr :
Pi :
PCtr :
PDtr :
PEtr :
PINDEX :
Pinv :
PLtr :
PM tr :
PV j :
Qtr :
rtr :
rl :
Intermediate consumption of good tr in activity j (volume)
12
12
Intermediate demand for good tr (volume)
3
Receipts from direct taxation on firms' income
1
Receipts from direct taxation on household h's income
4
Exchange rate
1
Equivalent variation for household h
4
Exports in good tr (volume)
3
Public expenditures
1
Investment demand for good tr (volume)
3
Total investment
1
Activity j demand for labour (volume)
4
Imports in good tr (volume)
3
Producer price of good i
4
Consumer price of composite good tr
3
Domestic price of good tr including taxes
3
Domestic price of exported good tr
3
GDP deflator
1
Price index of investment
1
Domestic price of good tr (excluding taxes)
3
Domestic price of imported good tr
3
Value added price for activity j
4
Demand for composite good tr (volume)
3
Rate of return to capital in activity tr
3
Rate of return to agricultural land
1
15
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
rc :
SF :
SG :
SH h :
TI tr :
TIEtr :
TIM tr :
VA j :
Rate of return to composite factor
1
w :
XStr :
YDH h :
YF :
YG :
YH h :
LS :
KDtr :
CAB :
Firms' savings
1
Government's savings
1
Household h's savings
4
Receipts from indirect tax on tr
3
Receipts from tax on export tr
3
Receipts from import duties tr
3
Value added for activity j (volume)
4
Wage rate
1
Output of activity tr (volume)
3
Household h's disposable income
4
Firms' income
1
Government's income
1
Household h's income
4
Total labour supply (volume)
1
Demand for capital in activity tr (volume)
3
Current account balance
1
Indtr ,t :
Demand for capital in activity tr (volume)
3
Ut :
Capital user cost
1
CtrMIN
,h
Minimum consumption of good tr by household h
:
12
Total: 151
Exogenous variables
Number of variables
DIVh :
DIV
ROW
:
LAND :
PWEtr :
PWM tr :
TGh :
XS NTR :
Dividends paid to household h
4
Dividends paid to the rest of the World
1
Land supply (volume)
1
World price of export tr
3
World price of import tr
3
Public transfers to household h
4
Output of activity NTR (volume)
1
Total: 17
Parameters
16
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
Production functions
Aj :
Scale coefficient (Cobb-Douglas production function)
aijtr , j :
Input-output coefficient
αj :
Elasticity (Cobb-Douglas production function)
io j :
Technical coefficient (Leontief production function)
vj :
Technical coefficient (Leontief production function)
CES function between capital and labor
AtrKL :
Scale coefficient
αtrKL
ρtrKL
σtrKL
:
Share parameter
:
Substitution parameter
:
Substitution elasticity
CES function between composite factor and land
AtrCL :
Scale coefficient
α Ctr L
:
Share parameter
ρCL
tr
σCL
tr
:
Substitution parameter
:
Substitution elasticity
CES function between imports and domestic production
AtrM :
Scale coefficient
αtrM :
Share parameter
ρtrM
σtrM
:
Substitution parameter
:
Substitution elasticity
CET function between domestic production and exports
BtrE :
Scale coefficient
βtrE
κtrE
τtrE
:
Share parameter
:
Transformation parameter
:
Transformation elasticity
LES consumption function
γ tr ,h :
Marginal share of good tr
Tax rates
tetr :
Tax on exports tr
17
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
tmtr :
txtr :
tyhh :
tyf :
Import duties on good tr
Tax rate on good tr
Direct tax rate on household h's income
Direct tax rate on firms' income
Other parameters
δj :
Share of activity j in total value added
λ hL :
Share of land income received by household h
λ
LF
:
Share of land income received by firms
λ LROW :
Share of land income received by foreigners
λ hR :
Share of capital income received by household h
λ
RF
:
Share of capital income received by firms
λ ROW :
Share of capital income received by foreigners
λW
h
Share of labor income received by household h
:
ψh :
µtr :
ng :
δ:
γ 1tr :
γ 2tr :
ir :
Propensity to save
Share of the value of good tr in total investment
Population growth rate
Capital depreciation rate
Parameter in the investment demand function
Parameter in the investment demand function
Real interest rate
sets
i, j ∈ I = { AGR,IND,SER,NTR }
tr ∈ TR = { AGR,IND,SER }
nag ∈ NAG = { IND,SER }
h ∈ H = { RP,UP,RR,UR }
t , t ∈ T = {1, 2,",10 }
All activities and goods (AGR: agriculture, IND: industry, SER:
services, NTR: non-tradable services)
Tradable activities and goods
Non-agricultural Tradable activities and goods
Households (RP: rural poor, UP: urban poor, RR: rural rich, UR:
urban rich)
Periods
18
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
Appendix II: GAMS Code
$TITLE
$STITLE
SEQUENTIAL DYNAMIC CGE MODEL FOR POVERTY ANALYSIS
EXTER+SD
$ontext
Model of a small open economy with government producing 4 goods using 3 factors owned by 4
households and recursive dynamics
THIS VERSION OF MAY,2004,
This copy could be used freely as long as proper reference is made to the source of the model
(c) COPYRIGHT: Nabil ANNABI, John COCKBURN AND Bernard DECALUWÉ
$offtext
*-------------------Variables definition---------------------------*
VARIABLES
*===============
*Prices
*===============
w(T)
r(TR,T)
rl(T)
* rf
rc(T)
P(I,T)
PD(TR,T)
PV(I,T)
PL(TR,T)
PC(TR,T)
PM(TR,T)
PE(TR,T)
PWM(TR,T)
PWE(TR,T)
PINDEX
PINV
e(T)
*===============
*Production
*===============
XS(I,T)
VA(I,T)
DI(TR,J,T)
CI(I,T)
Wage rate
Rate of return to capital in sector TR
Rate of return to agricultural land
Uniform return to capital
return to composite factor
Producer price of good I
Domestic price of good TR including tax
Value added price for sector I
Domestic price of good TR excluding tax
Price of composite good TR
Domestic price of imported good TR
Domestic price of exported good TR
World price of import TR (foreign currency)
World price of export TR (foreign currency)
Producer price index
Price index of investment
Exchange rate
Production of sector I
Value added in sector I (volume)
Intermediate consumption of good TR in sector J
Total intermediate consumption of sector I
*===============
*Factors
*===============
KD(TR,T)
LD(I,T)
LS(T)
LAND
CF(T)
*===============
*Demand
*===============
C(TR,H,T)
CTH(H,T)
Sector TR demand for capital
Sector I demand for labour
Total labour supply
Agricultural land
Composite agricultural capital-labor factor
Household H consumption of good TR (volume)
Household H total consumption (value)
19
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
INV(TR,T)
IND(TR,T)
IT(T)
ITVOL
DIT(TR,T)
G(T)
D(TR,T)
Q(TR,T)
Investment in good TR (volume)
INvestment by destination
Total investment (value)
Total investment (volume)
Intermediate demand for good TR
Total public consumption (value)
Demand for domestic good TR
Demand for composite good TR
*===============
*International trade
*===============
M(TR,T)
Imports of good TR
EX(TR,T)
Exports of good TR
CAB(T)
Current account balance
*===============
*Income and savings
*===============
YH(H,T)
YDH(H,T)
YF(T)
YG(T)
SH(H,T)
SF(T)
SG(T)
DIV(H,T)
DIV_ROW(T)
TG(H,T)
TI(TR,T)
TIM(TR,T)
TIE(TR,T)
DTH(H,T)
DTF(T)
nu(T)
adj(T)
savadj(T)
Household H income
Household H disposable income
Firms income
Government income
Household H savings
Firms savings
Government savings
Dividends paid to capitalist households
Dividends paid to foreigners
Public transfers to households
Receipts from indirect tax
Receipts from import duties
Receipts from tax on exports
Receipts from direct taxation on household H income
Receipts from direct taxation on firms income
Adjustment variable for hh savings
Adjustment variable for indirect taxes
Adjustment variable for investment and saving
*===============
*Others
*===============
EV(H,T)
LEON(T)
OMEGA
Equivalent variation for household H
Walras law verification variable
Objective variable
;
*-------------------------Equations definition------------------------*
EQUATIONS
*===============
*Production
*===============
SUPPLY(I,T)
VAD1(NAG,T)
VAD2(T)
VAD3(NTR,T)
ECF(T)
CIEQ(I,T)
DIEQ(TR,J,T)
LDEM1(TR,T)
LDEM2(NTR,T)
LANDEM(T)
Production function for sector I
Value added in non-agricultural sectors
Value added in agricultural sectors
Value added in non-tradable sectors
Composite agricultural labor-capital factor
Total intermediate consumption for sector I
Intermediate consumption of good TR by sector J
Labour demand for tradable sectors
Labour demand for non-tradable sectors
Agricultural land demand
*===============
*Income and savings
20
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
*===============
INCH(H,T)
INCDH(H,T)
INCF(T)
INCG(T)
SAVH(H,T)
SAVF(T)
SAVG(T)
*===============
*Taxes
*===============
Household income (workers)
Household H disposable income
Firms income
Government income
Household H savings
Firms savings
Government savings
INDTAX(TR,T)
IMDUTY(TR,T)
EXTAX(TR,T)
DIRTAXH(H,T)
DIRTAXF(T)
*===============
*Demand
*===============
Receipts
Receipts
Receipts
Receipts
Receipts
CTHEQ(H,T)
CONSH(TR,H,T)
CONSG(T)
INVEST(TR,T)
INVDF(TR,T)
INVVOL(T)
INTDEM(TR,T)
KACCUM(TR,T)
DEMOG(T)
from
from
from
from
from
indirect taxes on TR
import duties
tax on exports
household taxation
firm taxation
Household H total consumption
Household H consumption of good TR
Public consumption
Investment in good
Total investment in volume
Intermediate demand
*===============
*Prices
*===============
PRVA(I,T)
Value added price
RETK1(NAG,T)
Rate of return to capital from non-agricultural sectors
RETCF(T)
Return to composite capital-labor factor in agricultural sectors
RETK2(AGRS,T)
Rate of return to capital from agricultural sectors
PRDL(TR,T)
Domestic price
PRM(TR,T)
Import prices
PRE(TR,T)
Export prices
PRC(TR,T)
Composite price (tradables)
PRP(TR,T)
Producer price (tradables)
PII(T)
Price index for investment
AVPRI(T)
Producer price index
*===============
*International trade
*===============
CET(TR,T)
EXPORTS(TR,T)
ARMING(TR,T)
IMPORT(TR,T)
CURACC(T)
*===============
*Equilibrium
*===============
DOMABS(GOOD,T)
LEQUI(T)
INVDEM(T)
ISEQUI(T)
Relation between D and EX
Export supply
CES between imports and domestic good
Import demand
Current account
Domestic absorption (goods)
Labour market equilibrium
Investment-savings equilibrium
*===============
*Others
*===============
EVEQ(H,T)
WALRAS(T)
Calculation of EV
Verification of the Walras law
21
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
OBJ
Objective function
;
*===============
*Production
*===============
XS(I,T)
=E=
VA(I,T)/v(I);
VA(NAG,T)
=E= A_KL(NAG)*(alpha_kl(NAG)*LD(NAG,T)**(-rho_kl(NAG))
+(1-alpha_kl(NAG))*KD(NAG,T)**(-rho_kl(NAG)))
**(-1/rho_kl(NAG));
VA("AGR",T) =E= A_CL*(alpha_cl*CF(T)**(-rho_cl)+(1-alpha_cl)
*LAND(T)**(-rho_cl))**(-1/rho_cl);
CF(T) =E= A_KL("AGR")*(alpha_kl("AGR")*LD("AGR",T)**(-rho_kl("AGR"))
+(1-alpha_kl("AGR"))*KD("AGR",T)**(-rho_kl("AGR")))
**(-1/rho_kl("AGR"));
VA(NTR,T)
=E=
LD(NTR,T);
CI(I,T)
=E=
io(I)*XS(I,T);
DI(TR,J,T) =E=
LAND(T)
aij(TR,J)*CI(J,T);
=E= (((1-alpha_cl)*rc(T))/(alpha_cl*rl(T)))**sigma_cl*CF(T);
LD(TR,T)
=E= ((alpha_kl(TR)/(1-alpha_kl(TR)))**(sigma_kl(TR))
*(r(TR,T)/w(T))**sigma_kl(TR))*KD(TR,T);
LD(NTR,T)
=E= (P(NTR,T)*XS(NTR,T)-SUM(TR,DI(TR,NTR,T)*PC(TR,T)))/w(T);
*===============
*Income and savings
*===============
YH(H,T)
=E= lambda_w(H)*w(T)*SUM(I,LD(I,T))+lambda_r(H)
*SUM(TR,r(TR,T)*KD(TR,T))+lambda_l(H)*rl(T)*LAND(T)
+PINDEX(T)*TG(H,T)+DIV(H,T);
YDH(H,T)
=E=
YH(H,T) - DTH(H,T);
YF(T)
=E=
lambda_rf*SUM(TR,r(TR,T)*KD(TR,T))
+lambda_lf*rl(T)*LAND(T);
YG(T)
=E= SUM(TR,TI(TR,T))+SUM(H,DTH(H,T))+SUM(TR,TIE(TR,T))
+ SUM(TR,TIM(TR,T))+DTF(T)
;
SH(H,T)
=E=
nu(T)*psi(H)*YDH(H,T) ;
SF(T)
=E=
YF(T) - SUM(H,DIV(H,T)) - DTF(T) - e(T)*DIV_ROW(T) ;
SG(T)
=E=
YG(T) - G(T) - SUM(H,TG(H,T))*PINDEX(T);
*===============
*Taxes
*===============
TI(TR,T)
=E= (adj(T)+tx(TR)*(1+adj(T)))*(P(TR,T)*XS(TR,T)-PE(TR,T)*EX(TR,T))+(adj(T)
+tx(TR)*(1+adj(T)))*(1+tm(TR))*e(T)*PWM(TR,T)*M(TR,T);
TIM(TR,T)
=E=
tm(TR)*PWM(TR,T)*e(T)*M(TR,T);
TIE(TR,T)
=E=
te(TR)*PE(TR,T)*EX(TR,T);
DTH(H,T)
=E=
tyh(H)*YH(H,T)
DTF(T)
=E=
tyf*YF(T)
;
;
22
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
*===============
*Demand
*===============
CTH(H,T) =E= YDH(H,T)-SH(H,T);
C(TR,H,T)*PC(TR,T) =E= C_MIN(TR,H)*PC(TR,T)+gamma(TR,H)
*(CTH(H,T)-SUM(TRJ,C_MIN(TRJ,H)*PC(TRJ,T)));
G(T)
INV(TR,T)
=E=
XS("NTSER",T)* P("NTSER",T);
=E=
mu(TR)*IT(T)/PC(TR,T) ;
ITVOL(T)*PINV(T)
=E= IT(T);
IND(TR,T)/KD(TR,T) =E=
KD(TR,T+1)
DIT(TR,T)
SAVADJ(T)*SP(TR)*EXP(R(TR,T)/PINV(T));
=E= (1-DEP(TR))*KD(TR,T)+ IND(TR,T) ;
=E=
SUM(J, DI(TR,J,T)) ;
*===============
*Prices
*===============
PV(I,T)
=E= (P(I,T)*XS(I,T)-SUM(TR,DI(TR,I,T)*PC(TR,T)))/VA(I,T);
R(NAG,T)
=E= (PV(NAG,T)*VA(NAG,T) - w(T)*LD(NAG,T) )/KD(NAG,T) ;
R("AGR",T)*KD("AGR",T) =E= RC(T)*CF(T)-w(T)*LD("AGR",T);
RC(T)
=E=
(PV("AGR",T)*VA("AGR",T) - rl(T)*LAND(T))/CF(T);
PD(TR,T)
=E=
PL(TR,T)*(1+tx(TR))*(1+adj(T));
PM(TR,T)
=E=
(1+TX(TR))*(1+adj(T))*(1+tm(TR))*e(T)*PWM(TR,T);
PE(TR,T)
=E=
PWE(TR,T)*e(T)/(1+te(TR));
PC(TR,T)
=E=
(PD(TR,T)*D(TR,T)+PM(TR,T)*M(TR,T))/Q(TR,T);
P(TR,T)
=E=
(PL(TR,T)*D(TR,T) + PE(TR,T)*EX(TR,T))/XS(TR,T);
PINV(T)
=E=
PROD(TR$(INVO(TR) NE 0), (PC(TR,T)/mu(TR))**mu(TR));
PINDEX(T)
=E=
SUM(I,PV(I,T)*delta(I));
*===============
*International trade
*===============
XS(TR,T)
=E= B_E(TR)*(beta_e(TR)*EX(TR,T)**kappa_e(TR)
+(1-beta_e(TR))*D(TR,T)**kappa_e(TR))**(1/kappa_e(TR));
EX(TR,T)
=E=
Q(TR,T)
=E= A_M(TR)*(alpha_m(TR)*M(TR,T)**(-rho_m(TR))
+(1-alpha_m(TR))*D(TR,T)**(-rho_m(TR)))**(-1/rho_m(TR));
M(TR,T)
=E= ((alpha_m(TR)/(1-alpha_m(TR)))**(sigma_m(TR))
*(PD(TR,T)/PM(TR,T))**sigma_m(TR))*D(TR,T);
((PE(TR,T)/PL(TR,T))**tau_e(TR)*((1-beta_e(TR))/beta_e(TR))
**tau_e(TR))*D(TR,T);
CAB(T)
=E= SUM(TR,PWM(TR,T)*M(TR,T))+ lambda_row*SUM(TR,r(TR,T)*KD(TR,T))/e(T)
+lambda_lrw*rl(T)*LAND(T)/e(T)+DIV_ROW(T)-SUM(TR,PWE(TR,T)*EX(TR,T));
*===============
*Equilibrium
*===============
Q(GOOD,T) =E=
SUM(H,C(GOOD,H,T))+DIT(GOOD,T)+INV(GOOD,T);
23
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
LS(T)
LS(T+1)
=E=
=E=
SUM(I,LD(I,T));
(1+GN)* LS(T) ;
IT(T)
=E=
PINV(T)*SUM(TR,IND(TR,T)) ;
IT(T)
=E=
SUM(H,SH(H,T)) + SF(T) + SG(T) + CAB(T)*e(T);
*===============
*Others
*===============
EV(H,T)
LEON(T)
=E=
PROD(TR,(PCO(TR)/PC(TR,T))**gamma(TR,H))
*(CTH(H,T)-SUM(TRJ,C_MIN(TRJ,H)*PC(TRJ,T)))
-(CTHO(H)-SUM(TRJ,C_MIN(TRJ,H)*PCO(TRJ)));
=E=
Q("SER",T)- SUM(H,C("SER",H,T))-DIT("SER",T);
OBJ..
OMEGA
=E=
1;
*------------------------Initialization----------------------------*
w.L(T)
r.L(TR,T)
rl.l(T)
rc.l(T)
P.L(I,T)
PD.L(TR,T)
PV.L(I,T)
PL.L(TR,T)
PC.L(TR,T)
PM.L(TR,T)
PE.L(TR,T)
PWM.L(TR,T)
PWE.L(TR,T)
PINDEX.L(T)
PINV.L(T)
e.L(T)
XS.L(I,T)
VA.L(I,T)
DI.L(TR,J,T)
CI.L(I,T)
KD.L(TR,T)
LD.L(I,T)
LS.L(T)
LAND.L(T)
CF.L(T)
C.L(TR,H,T)
CTH.L(H,T)
INV.L(TR,T)
IT.L(T)
ITVOL.L(T)
DIT.L(TR,T)
G.L(T)
D.L(TR,T)
Q.L(TR,T)
M.L(TR,T)
EX.L(TR,T)
CAB.L(T)
YH.L(H,T)
YDH.L(H,T)
YF.L(T)
YG.L(T)
SH.L(H,T)
SF.L(T)
SG.L(T)
DIV.L(H,T)
DIV_ROW.L(T)
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
wo;
ro(TR);
rlo;
rco;
PO(I);
PDO(TR);
PVO(I);
PLO(TR);
PCO(TR);
PMO(TR);
PEO(TR);
PWMO(TR);
PWEO(TR);
PINDEXO;
PINVO;
eO;
XSO(I);
VAO(I);
DIO(TR,J);
CIO(I);
KDO(TR);
LDO(I);
LSO;
LANDO;
CFO;
CO(TR,H);
CTHO(H);
INVO(TR);
ITO;
ITVOLO;
DITO(TR);
GO;
DO(TR);
QO(TR);
MO(TR);
EXO(TR);
CABO;
YHO(H);
YDHO(H);
YFO;
YGO ;
SHO(H);
SFO;
SGO;
DIVO(H);
DIV_ROWO;
24
Annabi, N. Cockburn, J and Decaluwé, B. (2004)
TG.L(H,T)
TI.L(TR,T)
TIM.L(TR,T)
TIE.L(TR,T)
DTH.L(H,T)
DTF.L(T)
nu.FX(T)
adj.L(T)
*OTHERS
LEON.L(T)
OMEGA.L
=
=
=
=
=
=
TGO(H);
TIO(TR);
TIMO(TR);
TIEO(TR);
DTHO(H);
DTFO;
= 1;
= 0;
=
=
0;
1;
*----------------------------Closure--------------------------------*
* Exchange rate is the numeraire, capital is sector specific, fixed public
* expenditure (in volume).
LAND.FX(T)
= LANDO;
ITVOL.L(T)
= ITO/PINVO;
DIV.FX(H,T)
= DIVO(H);
DIV_ROW.FX(T)
= DIV_ROWO;
TG.FX(H,T)
= TGO(H);
XS.FX("NTSER",T) = XSO("NTSER");
PWM.FX(TR,T)
= PWMO(TR);
PWE.FX(TR,T)
= PWEO(TR);
e.FX(T)
= eo;
CAB.FX(T)
= CABO;
SG.FX(T)
= SGO;
adj.L(T)
= 0;
*---------------------------savadj.L(t)
= 1 ;
LS.L(T)
= LSO;
KD.L(TR,T)
= KDO(TR);
IND.L(TR,T)
= INDO(TR) ;
LS.FX(T1)
KD.FX(TR,T1)
= LSO;
= KDO(TR);
* Simulation
*tm(TR)
= 0;
*---------------------------Model execution---------------------------*
OPTION NLP=CONOPT3;
*OPTION NLP= MINOS5;
option limrow=1;
MODEL EXTERPSD OPEN ECONOMY WITH GOVERNMENT
EXTERPSD.HOLDFIXED = 1 ;
EXTERPSD.WORKSPACE = 100 ;
SOLVE EXTERPSD MAXIMIZING OMEGA USING NLP;
/ALL/;
25