Model Equations
Complement to the Article submitted to the Journal of Policy Modeling Titled
The Euro-Mediterranean Free Trade Agreement
An Inquiry into the Cost of Adjustment to Tariff Liberalization for the Egyptian Economy
An Intertemporal General Equilibrium Analysis
Abeer Elshennawy
Assistant Professor
The American University in Cairo
AUC Avenue, P.O. Box74. New Cairo 11835
Egypt
e-mail abeershennawy@yahoo.com
ashenawy@aucegypt.edu
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Model Equations
The model draws upon the contributions to intertemporal General Equilibrium models by
Mercenier and Sampaio de Souza (1993), Mercenier (1995), Go (1991) , Diao and Somwaru
(1996), Diao and Somwaru (1997), Diao and Somwaru (1999) and is composed of two parts: a
dynamic part - in which both households and firms decision to consume and invest is a result of
dynamic optimization- and a standard within period static CGE model. A detailed description of
the dynamic part can be found in Elshennawy, 2011
The consumer problem:
The representative household chooses the path of consumption that maximizes the discrete
intertemporal utility function :
Uo  t (1/1)t ln(TCt )
A1
The intertemporal budget constraint is
t=1Rt PtctTCt  t=1Rt ( wl p,t LSUP p,t + wl NP,t LSUP NP,t +THGt )+
A2
TCt   s CDSa,(ts )
A3
Rt =  =0
1
1+r
A4
Variables and Parameters
= Aggregate consumption
TCt
CD( S ,t )
= private consumption of good s
Rt
Ptct
= discount factor from time t to time zero
wl( P ,t )
wl( NP ,t )
LSUP( P ,t )
= wage rate of production labor
= the price of total consumption
= wage rate of nonproduction labor
= labor supply of production labor
1
LSUP( NP ,t ) = labor supply of nonproduction labor
THGt
rt
 
= transfer of government revenue to household
= instantaneous interest rate
= value of the household initial financial wealth
SAVt = wl p,t LSUPp,t +wlNP,t LSUPNP,t + s DIVs +THGt - rt Dt 1 - PtctTCt
A5
First order (Euler conditions) imply
TCt 1
Ptct
(1  ) 
(1 rt )
TCt
Ptct 1
A6
The Firm Problem
Asset market equilibrium condition
r
where
DIVs Vs

Vs
Vs
r
is the world interest rate,
A7
DIV is dividends, V is the value of the firm. In addition,
the following terminal condition is imposed to rule out Ponzi schemes
limt  RV
t S ,t  0
A8
solving the above difference equation yields

VS ,1  t 1 Rt DIVS ,t
A9
The market value of the firm is defined as the sum of discounted stream of future
dividends. Dividends are defined as
2
PVA f[(L ,K ] - wl P, t LSUP P, t - wl NP, t LSUP NP, t - ADC - PI I
S,t
S,t S,t
S,t
S ,t S ,t
ADCS ,t
I S2,t
 S PVAS ,t
K S ,t
A11
A10
and
ADC is the adjustment cost of investment while PVA is value added price
Capital accumulation constraint
K S ,t 1  (1   S ) K S ,t  I S ,t
A12
Differentiating with respect to the control variable
qS ,t  PI S ,t  2PVAS ,tS
I
yields
I S ,t
KS ,t
A13
which determines the shadow price of capital (Tobin q).
Differentiating with respect to the state variable K yields the no arbitrage condition
wkS ,t  PVAS ,tS (
I S ,t 2
)  (1   S )qS ,t  (1  r )qS ,t 1  0
K S ,t
which is the same as the asset equilibrium condition since V=q K.
Variables and Parameters
wk( S ,t )
=capital rental rate
PVA( S ,t )
=value added price
3
A14
K ( S ,t )
S 
=capital stock

=depreciation rate

I S ,t  AK S  S  INVDS S ,S,S
A15
is a composite good produced from all final goods - with fixed share  - using a constant
returns technology,
It
Variables and Parameters
=shift parameter in the investment function
AK S
INVD( S  ,S )
=investment demand by sector of origin
I ( S ,t )
PI ( S ,t )
=new physical capital good
=price of new capital good
Current Account Dynamics
In open economy, investment is not constrained by the availability of domestic savings
and any discrepancy is financed through foreign borrowing. Current account equilibrium is
described by
Dt  Dt 1  rt Dt 1  TBt
A16
The above equation shows that the increase in foreign debt
deficit
TBt
Dt
is composed of the trade
and the interest payment on foreign debt .
Within period equations (the time subscript is omitted to simplify notation)
Price block
Domestic price of imports
PM S ,R  [1  mS ,R ]PWIM S ,R
A17
4
Domestic price of exports
PES ,R  [1   eS ,R ]PWES ,R
A18
Composite import price
m
1
1
m
1+ m
1
m
PMM S =
[  m1+ m PM S1+,W m + (1  )m1+ m PM S1+,EU
] m
ACM
A19
Composite export price
E
E
E 1
1
1  -  1-1 E 1
E 1
PEES =
[ E E PES ,W +(1-  E )- E 1PES,W
] E
ATE
A20
Domestic supply price
PCS  PDS
DS
MM S
 PMM S
CS
CS
A21
Domestic output price
PX S  PDS
DS
EES
 PEES
XS
XS
A22
Value added price
PVAS  [1  iS ]PX S   S  IOS  ,S PCS 
A23
Price of investment (unit cost function for producing investment good)
5
PI S 


S
PCS S  ,S
AK S . S  
A24

S  ,S
S  ,S
Variables and Paramters:
Variables
= domestic good price (sold domestically)
PDS
PM ( S , R )
= domestic price of imports from region R
PMM S
= composite price of imports
PE( S , R )
= domestic price of exports to region R
PEES
PCS
PX S
PVAS
DS
= composite price of exports
M ( S ,R )
= quantity of imports from region R
= domestic supply price
= domestic output price
= value added price
= quantity of domestic output sold domestically
= composite quantity of imports
MM S
= quantity of exports to region R
E( S ,R )
= composite quantity of exports
EES
CS = quantity of good supplied domestically
X S = quantity of domestic output
Parameters
 mS  
 eS 
 iS 
= import tariff rate

= export subsidy rate

= input output coefficient
PWIM S , R
PWES ,R
= world price of imports from region R
( S ,S ) 
= share of good s’ in sector s investment
AK S
= shift parameter in the investment production function

= world price of exports from region R
6
Output supply and demand block


AX S [KS SK LSSL ]
A25
Labor is a Cobb Douglas aggregation of two skill categories; production and nonproduction
labor


LS  LS S LS S
P
P
A26
NP
NP
Profit maximization by producers lead to the following factor demand equations
Factor demand
LDS 
P
LDS
NP
PVAS X S S  S
L
wlP
P
PVAS X S S  S
L

wlNP
A27
NP
A28
The rental rate of capital is determined from the following equation
KS 
PVAS  K X S
wkS
A29
Intermediate demand
INTDS   S  IO( S ,S  ) X S 
A30
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Intermediate demand is determined according to leontif technology and is therefore equal
to the sum of fixed input output coefficients
IO( S ,S  )
multiplied by domestic output
X S
Armington Functions
CS  ACS [S MM SCS  (1 S )DSCS ]1/ CS
MM S 
ACm[ m M S,Wm
 1
 m
 (1   m )M S ,EU ] m
A31
A32
Import demand
MM S  DS [
PDS S
]1/(1CS )
PMM S .(1 S )
M ( S ,W ) = M (S,EU) [
A33
PM (S,EU)  m
1
] 1+ m
PM (S,W) (1-  m )
A34
CET Functions
X S  ATS [ S EESTS  (1  S )DSTS ]1/ TS
A35
1
EES = ATE[  E ES,EW +(1-  E )ES,EEU ] E
8
A36
Export Supply
EES  DS [
PEES 1  S ) 1/( S 1)
]
PDS  S
E(S,W) = E(S,EU) [
A37
PE(S,W)  E
1
] ( E -1)
PE(S,EU) (1-  E )
A38
Since there are no imports in the case of nontradables, supply consists only of output
sold domestically.
Variables and Parameters:
wlP = production labor wage rate
wlNP = non production labor wage rate
wkS = capital wage rate
KS = capital stock in sector s
LD( P,S ) =demand for production labor by sector s
LD( NP,S ) = demand for non production labor by sector s
INTDS =intermediate demand of good S
AX S = production function shift parameter
( K ,S ) = share of capital in value added of sector s
 ( L,S ) = share of labor in value added of sector s
 ( P,S ) =share of production labor in total labor input
( NP,S ) =share of non production labor in total labor input
ACS = shift parameter in Armington
ACM S = shift parameter in Regional Armington elasticity
S = share parameter in Armington
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 (m,S ) = share parameter in Regional Armington
CS = Armington exponent
(m,S ) = Regional Armington exponent
ATS = shift parameter in CET
ATES = shift parameter in Regional CET
 S = share parameter in CET
 E = share parameter in Regional CET
T = CET exponent
S
 E = Regional CET exponent
 CS = elasticity of substitution between domestic goods and imports
 X S = elasticity of substitution between domestic use and exports
Factor and institution block
Household flow income
YHt  wl( P,t ) LSUP( P,t )  wl( NP,t ) LSUP( NP,t )  S DIV(S ,t )  THGt  rt Dt 1
A39
Household demand
PCS CDS  aS (YH  SAV )
Demand for each good is a fixed share
A40
a
of total expenditure on goods
Sectoral investment demand (by sector of origin)
PCS  INVD( S  ,S )  ( S  ,S ) PI S .I S
A41
Variables and Parameters
YH = household income
D = foreign debt
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SAV = household saving
CDS =HH demand for good S
INVD( S  ,S ) = sector S investment demand for good S’ (investment demand by sector of origin)
I S = new investment
aS = spending share for HH on good S
AKS = shift parameter in investment equation
( S  ,S ) = share of good s’ in sector s investment
Government transfers
 iS PX S X S +  R s mS,R PWIM S,R M S,R -  R s eS ,R PWES,R ES,R
A42
Government transfers to household THG is a transfer of net government revenue which
consists of government income that is revenue from indirect taxes and tariffs less subsidies.
Government consumption and government deficit is ignored. Consequently, household savings
is equal to national savings.
Equilibrium conditions
Factor market equilibrium
Labor
 S S  S
L
 S S  S
L
p
PVAS X S  LSUPp wLp
NP
A43
PVAS X S  LSUPNP wlNP
A44
Demand for labor of each skill category equal its supply.
Capital
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 S S .PVAS .X S  K S .wkS
A45
k
Commodity market equilibrium
CS =CDS +INVDS +INTDS
Supply of output for good S,
A46
C must equal consumption demand, investment demand and
intermediated demand.
Current account
SPWIM S M S - SPWES ES =TB
A47
current account equilibrium requires imports less exports must equal foreign savings
Variables and Parameters
KS =capital supply (capital stock)
TB = Foreign savings
LSUPP = supply of production labor
LSUPNP = supply of non production labor
 d = direct tax rate
Terminal condition (last period is SS)
 KS S = ISS
A48
The first condition indicates that in the steady state investment is equal to depreciated capital.
rSS = 
A49
The second condition follows form the Euler equation evaluated in the steady state and states
that the interest rate must be equal to the rate of time preference. Assuming that the world
economy is in a steady state and therefore (r )is constant implies in turn that the rate of time
preference is also constant.
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rSS DSS +TBSS =0
A50
Debt is constant in the steady state. If
D
is positive then foreign savings (the trade deficit)
must be negative. That is the country must run a trade surplus in order to pay the interest rate on
debt. If the above equation holds then domestic savings must be equal to investment.
rSSVSS = DIVSS
A51
Since the condition for asset market equilibrium and the no arbitrage condition are
equivalent, the above equation must hold in the steady state implying that the average return to
capital is constant.
Variables and Parameters:
TBt = foreign savings ( trade deficit)
r = interest rate
Welfare evaluation (Equivalent Variation index)
t

 1 


t 0  1   

 
ln  TC

   1 



 t 0  1   
1   
t
ln TCt
A52

Where
TC
is base year consumption. This implies that the welfare gain arising for a policy
change is equivalent form the point of view of the representative household to increasing the
reference consumption profile by

REFERENCES
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A Global Intertemporal General Equilibrium Analysis”, Department of Applied Economics.
University of Minnesota.
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Equilibrium Evaluation of the Regional Integration”, International Economic Journal. Vol 13,
No1, pp.27-43.
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General Equilibrium Model for Egypt”, Econmodels.com. Elsevier.
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Forward-Looking CGE Model of the Philippines”, Journal of Development Economics, Vol 44.
pp. 229-261.
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Indebted Market Economy: Brazil”, In J. Mercenier and T. Srinivasan eds. Applied General
Analysis and Economic Development, Ann Arbor. Univesity of Michigan Press.
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