AN INTRODUCTION TO APPLIED GENERAL
EQUILIBRIUM (AGE) MODELING
by
Paul de Boer*
Cristina Mohora
Frédéric Dramais
* [email protected] www.ecomod.net
Prepared for: the Evening Course on Monday (19:00-21:00, Room RM-120), 14th
International Conference on Input-Output Techniques, 10-15 October 2002, Montréal,
Canada.
Date : September 2002
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2
1.
An introduction to applied general
equilibrium (AGE) modeling
Suppose that you are the economic advisor to your government that is involved in
90negotiations for acceding to the WTO. The government’s budget is balanced and the
trade balance is in equilibrium. Your country has to reduce its tariffs on all imported
commodities by 30%. This would imply a loss of revenues and your government asks you by
which percentage it has to increase the taxes on private consumption of commodities so as
to compensate for this loss. It also wants to know the impact of this policy measure on a
number of important economic variables, such as unemployment.
How does an AGE modeler proceed?
In the example that we will deal with, there are five economic actors: two firms, one
household, a bank that allocates savings over investments, the government and the rest of
the world.
In a Social Accounting Matrix (SAM) the transactions between these actors are recorded, so
that the SAM forms the basis for (static) AGE modeling. It is assumed that it is the
benchmark equilibrium of the economy.
The modeler constructs a model in which he describes the behavior of each of the economic
actors and chooses the parameters of the model in such a way that the benchmark
equilibrium is reproduced (“calibration” of parameters).
The modeler decreases the value of the tariff rates by 30% and solves his model to obtain
the values that the tax rates on private consumption of commodities have to assume in
order to compensate for the loss in revenues. He assesses the impact of this policy change
by comparing the implied unemployment with the unemployment in the benchmark
equilibrium.
In the lecture we will summarize these documents, and give examples of simulations with an
AGE model for Romania. We will be delighted to run any simulation at your request.
____________________________________________________________
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3
1.
The Example
Build a general equilibrium model for the economy described below.
Actors:
1. Firms
There are two firms that produce one domestic commodity each, maximize their profits and
face a nested production function in capital, labor, and intermediates.
The domestically produced commodity is allocated over the domestic market and exports
according to a constant elasticity of transformation (CET) function.
The domestically produced commodity delivered to the domestic market and the imports are
combined into a composite commodity by means of a constant elasticity of substitution
function (ARMINGTON assumption).
Capital and labor are mobile among sectors, so that the returns to capital and to labor are
the same for both firms.
2. Household
There is one household that owns the capital and that offers labor.
The time endowment is fixed at 400 and the capital endowment at 170.
Its savings are a fixed fraction of net income, i.e. the income after payment of income tax.
It maximizes a so-called Extended LES utility function with the two consumption
commodities and leisure as arguments, subject to its budget constraint.
The price of leisure is the net wage rate, i.e. the wage rate after payment of income tax.
On the labor market there is a negative relationship between the rate of change in real
wages, and the rate of change in the unemployment rate. The consumer price index is of the
Laspeyres type; the initial unemployment is equal to 10.
The replacement rate (the percentage of the wage rate that an unemployed person
receives) is equal to 50%. The other transfers to the household are constant in real terms,
and equal to 15. In order to make them nominal we use the (Laspeyres) consumer price
index.
3. Bank
Savings are allocated over the investment demand of the two firms according to a CobbDouglas utility function.
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4
4. Government
The government maximizes a Cobb-Douglas utility function with the government
consumption of the two commodities, capital and labor as arguments, under a balanced
budget.
Taxes are proportional to their tax bases.
5. Rest of the World
The import prices are exogenously given ("small economy assumption"); the balance of
payments is in equilibrium.
Database for the economy:
Sec1
Sec2
K
TRK
L
TRL
Total
Sec1
5
15
60
12
20
8
120
Sec2
40
20
50
10
90
36
246
Imports
Tariffs
16
8
30
10
Gov
20
50
60
0
120
0
250
Cons Invest
64
5
150 15
Exports
10
36
Tax on consumption
: of commodity 1
: of commodity 2
:5
: 37
Tax on capital
: of sector 1
: of sector 2
: 12
: 10
Tax on labor
: of sector 1
: of sector 2
:8
: 36
Tariffs on imports
: of sector 1
: of sector 2
:8
: 10
Income tax
Total tax revenues
Transfer of unemployment benefits
Other transfers to the household
: 144
--------+
: 270
:5
:15
____________________________________________________________
Copyright Ecomod Network, not to be disclosed to third parties without prior written approval
EcoMod Network – ULB – 50, av. F. Roosevelt – CP 140 – B-1050 Brussels – Belgium
Phone: +32 2 650 39 88 – Fax: +32 2 650 41 37 – E-mail: [email protected]
Total
144
286
5
2.
Notation
In the forthcoming documents we will use the notation that is summarized below. The
expression “(sec)” is an index: sec = 1,2. It refers to the pertinent firm (or sector). It is used
in the software package GAMS that is widely used for solving large-scale AGE models.
Variables:
C(sec)
CBUD
CEBUD
CG(sec)
CV
: demand of consumer commodities by the household
: consumption budget of the household
: extended budget of the household
: demand of consumer commodities by government
: compensating variation
E(sec)
ER
EV
: export of the domestically produced commodities
: exchange rate
: equivalent variation
I(sec)
: demand for investment commodities
KG
KS
K(sec)
: capital use of government
: capital endowment
: capital demand by firms
LG
LS
L(sec)
: labor use of government
: labor endowment
: labor demand by firms
M(sec)
: imports of commodities
P(sec)
PCINDEX
PD(sec)
PE(sec)
PK
PL
PM(sec)
PWEZ(sec)
PWMZ(sec)
PROFIT(sec)
: price of composite commodities
: Laspeyres consumer price index
: price of domestically produced commodities
: export price (in local currency)
: return to capital
: wage rate
: import price (in local currency)
: world price of exports
: world price of imports
: profits by firms
S
SF
SG
SH
: total savings
: foreign savings
: government savings
: household savings
TAXR
TRC(sec)
TRF
TRICK
TRK(sec)
: total tax revenues
: tax revenues from consumer commodities
: total transfers
: artificial objective variable
: tax revenues from capital use
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6
TRL(sec)
TRM(sec)
TRO
TRY
TS
: tax revenues from labor use
: tax revenue on imports
: other transfers to the household
: tax revenues from household's income
: time endowment
U
UNEMP
: utility level of the household
: unemployment
X(sec)
X(sec,sec)
XD(sec)
XDD(sec)
: composite commodity
: inter-industry deliveries
: supply of domestically produced commodities by firms
: domestic commodity supplied to the domestic market
Y
: household's total income
Parameters:
aA(sec)
aF(sec)
aT(sec)
: efficiency parameter of the Armington function
: efficiency parameter in the firm's production function
: efficiency parameter of the CET function
CG(sec)
HLES(sec)
I(sec)
KG
LG
: Cobb-Douglas power of commodities bought by government
: marginal budget shares of the household’s LES utility function
: Cobb-Douglas power of the bank's utility function
: Cobb-Douglas power of capital use by government
: Cobb-Douglas power of labor use by government
A(sec)
F(sec)
T(sec)
: share parameter of the imports in the Armington function
: share parameter of K of the firm's CES production function.
: share parameter of exports in the CET function
io(sec,secc)
: technical coefficients of the inter-industry flows
mps
H(sec)
: marginal propensity to save
: subsistence level
nif
phillips
: net income fraction
: value of the Phillips parameter
A(sec)
F(sec)
T(sec)
: elasticity of substitution of the Armington function
: elasticity of substitution between K and L in the CES function
: elasticity of transformation of the CET function
tc(sec)
tk(sec)
tl(sec)
tm(sec)
trep
ty
: tax rate on consumer commodities
: tax rate on capital use
: tax rate on labor use
: tariff rate
: replacement rate
: tax rate on income
____________________________________________________________
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7
3.
The Social Accounting Matrix (SAM)
When constructing a general equilibrium model one often needs to collect data from
different statistical sources. Then, one obtains a database that is similar to the one that we
have given in the example.
In applied general equilibrium modeling the database is usually presented in the form of a
so-called Social Accounting Matrix (SAM).
In a SAM all transactions in an economy are described:
1. between domestic agents (in our model between the household, the two firms, the
bank and the government); and
2. between domestic agents and the Rest of the World.
In table 1 we give the SAM that corresponds to the database of the example. As base year
for the price indices we take the year to which the SAM refers, so that we put them all equal
to 1. Therefore, the nominal values of the elements are equal to the real ones.
Table 1. A (stylized) SAM
Account type
SubDivision
com 1
Commodities
com 2
sec 1
Producing sectors
sec 2
capital
Primary income
labor
household
Household
taxes/subs
Government
Savings/investment savings
Rest of world (RoW) imports
Total supply
Commodities
com 1 com2
1
1
110
2
210
3
4 5 8
10
6 7 16
30
8 134 250
Sectors Primary income
Expenditure
RoW
sec1 sec2 capital labor househ. governm. investm. exports
2
3
4
5
6
7
5
40
64
20
5
15 20
150
50
15
10
36
60 50
60
20 90
120
170
230
20
20 46
186
20
120 246
170
230
420
270
20
46
Let us first consider the account type “Commodities” which, in the example, consists of the
composite commodities produced by the two firms. Row wise we have the domestic sales
(destination) of the composite commodities: on the one hand to “Sectors”, that produce the
composite commodities, and, on the other one, to final users comprised in “Expenditure”.
In the block (Commodities, Sectors) we have the intermediate demand ( X ij ) and in the
block (Commodities, Expenditure) the final demand ( Ci , CG i , I i ). In the final column (Total)
we have the total domestic sales of the two composite commodities ( X i ).
Column wise, we have the origin of the “Commodities”.
The imports are composed of two components: the value c.i.f. ( M i ), supplied by the “Rest
of the World” and the import duties levied by the “Government”.
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© Ecomod Network
Total
8
134
250
120
246
170
230
420
270
20
46
8
Consider commodity 1. The total supply (the column sum) is equal to the total sales (the row
sum) of 134. In the block (Rest of the World, Commodities) we have the imports c.i.f. (for
commodity 1 equal to 16) and in the block (Government, Commodities) the import duties (for
commodity 1 they are equal to 8).
Consequently, it follows that the domestically produced commodity 1 which is delivered to
the domestic market ( XDD 1 ) is equal to:
134 - 16 - 8 = 110
Similarly, we derive that the domestically produced commodity 2 which is delivered to the
domestic market is equal to 210.
These are the figures in the block (Sectors, Commodities).
Next, we turn to the account type “Sectors”.
Row wise we have the sales of the domestically produced commodity to the domestic
market ( XDD i ), which have already been dealt with above, and to the foreign market, i.e.
the exports f.o.b. ( E i ) recorded in the block (Sectors, Rest of the World). As row totals we
have the supply of domestically produced commodities ( XD i ), 120 and 246, respectively.
Column wise we have the inputs required for the production of the domestically produced
commodity. They consist of intermediates that, as we have seen above, are recorded in the
block (Commodities, Sectors), and value added that, in this simple framework, is equal to
the remunerations of the factors of production capital ( K i ) and labor ( L i ). In the block
(Primary income, Sectors) we find these components of value added. In the block
(Government, Sectors) we find the taxes on capital and labor (for sector 1 they are equal to
12 and 8, respectively, the total being 20).
In the third account, Primary Income, we have Row wise two blocks with non-zero values.
The first block, (Primary income, Sectors), has already been dealt with, and in the second
block, (Primary income, Government), the remunerations for the use of capital and labor by
the government are recorded (60 and 120, respectively). The row totals are 170 for capital
and 230 for labor. These totals are found back Column wise in the block (Household
expenditures, Primary income).
In the account “Household expenditure” we have Row wise the resources that are at the
disposal of the household, viz. its revenues from capital and labor and, as secondary
income, the transfers from the government (20), which, in the example, consist of
unemployment benefits (5) and other transfers (15).
Column wise we see how the household spends its resources. In the block (Commodities,
Household) we find the expenditures on the consumption of commodities (64 on commodity
1 and 150 on commodity 2). In the block (Government, Household) we record all taxes paid
by the household: taxes on consumption (5 and 37, respectively) and the income tax of 144,
so that the total taxes paid are equal to 186. Finally, in the block (Savings, Household) we
record the household’s savings of 20.
Since in our example the government budget is balanced and the balance of payments is in
equilibrium, the government and foreign savings are equal to zero, so that the accounts
“Savings/investment” and “Rest of the World” have, implicitly, been dealt with in the
discussion above.
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© Ecomod Network
9
4.
Actor 1: The Firms
In figure 1 we summarize the production structure:
Figure1. Production of the domestic commodity, domestic supply, production
of the composite commodity and domestic demand
X11
K1
X21
K2
L1
L2
X12
X22
Leontief
CES
CES
Leontief
Intermediates
sector 1
Value added
sector 1
Value added
sector 2
Intermediates
sector 2
Leontief
Leontief
XD1
XD2
CET
CET
E1
XDD1
X12
XDD2
Armington
Armington
X1
X2
Intermediate
demand
X11
M2
M1
Final
demand
C1
CG1
Final
demand
I1
C2
CG2
E2
Intermediate
demand
I2
X21
X22
Now, we discuss the relations between the variables for firm 1 only and show where
to find the figures in the SAM. The ensuing equations will be given in the forthcoming
document: Model.doc.
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© Ecomod Network
10
For the production of the domestically produced commodity, XD1 , we use a twolevel production function. At the first level “intermediates” and “value added” are
aggregated to XD1 by means of a Leontief production function (“no substitution”). At
the second level we have on the one hand the inter-industry flows, aggregated by a
Leontief function to “intermediates”, and on the other hand capital and labor that are
aggregated by means of a Constant Elasticities of Substitution function (CES) to
“value added”. The firm is assumed to minimize its costs when producing XD1 .
From the second column of the SAM (Sectors) we gather that in the benchmark
equilibrium, where nominal and real values are the same due to the fact that we have
put all price indices equal to one, that:
X11  5
X 21  15
K1  60
L1  20 and
XD1  120
The domestically produced commodity is either supplied to the domestic market,
XDD 1 , or to the foreign market (exports, E1 ). They are not perfect substitutes. We
assume that the firm maximizes its revenues from supplying to the domestic market
and to the foreign market subject to a transformation function that has the same
mathematical structure as the CES function, but which is referred to as the CET
(Constant Elasticity of Transformation) function.
In the second row of the SAM (Producing sectors) we have:
XDD1  110
E1  10
and
XD1  120
For the production of commodities for intermediate and final demand the firm can use
commodities from domestic origin, i.e. XDD1 and from foreign origin, i.e. the imports
M1 .They are not perfect substitutes (the famous “Armington assumption”) so that the
firm is assumed to minimize its costs when producing the composite commodity X1 .
In the first column of the SAM (Commodities) we have:
XDD1  110
Import tariffs = 8
M1  16
and
X1  134
This composite commodity is delivered to intermediate demand, X11 and X12 , and to
the final uses: the demand by the household, C1 , the demand by government, CG1 ,
and investment demand, I 1 .
Finally, in the first row (Commodities) we find:
X11  5
X12  40
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© Ecomod Network
C1  64
CG1  20
I1  5
and
X1  134
11
5.
Actor 2: The Household
In figure 2 we summarize the decisions of the Household.
Figure 2. The decisions of the Household
TRO
KS
TS
Leisure (C3)
LS
Income from
capital
Income from
transfer
UNEMP
Labor
demand
(LS-UNEMP)
Unemployment
benefits
Income from
labor
Income (Y)
Savings
Taxes
Income
from leisure
CBUD
CEBUD
Expenditure on
commodity 1
Expenditure on
commodity 2
In the example we assume that the capital endowment (KS), the time endowment
(TS) and the other transfers (TRO) from the government, such as the payment of
pensions, are exogenously given.
The household maximizes its utility in two stages: in the first one it allocates its time
endowment over labor supply (LS) and leisure (the “consumption commodity” 3, C 3 ).
We allow for unemployment so that the labor demand is smaller than the labor
supply. The household receives unemployment benefits (at 50% of the wage rate)
_____________
© Ecomod Network
12
and income from labor. All four sources of income together yield the household
income (Y).
It pays income taxes and saves a fixed fraction out of its net income. Subtracting
taxes and savings from income yields the budget (CBUD) that it devotes to the
purchase of commodities 1 and 2. In the second stage the consumer maximizes a
utility function, with the consumption of these commodities as arguments, subject to
its budget constraint.
For both stages we use a Linear Expenditure System (LES). This is equivalent to the
maximization of an Extended LES utility function, with the consumption of the two
commodities and of leisure as arguments, subject to the extended budget, in which
the income for leisure is included (CEBUD).
In the third row and third column of the SAM (Primary income) we find the income
from capital (170) and from labor (230), whereas in the fourth row (Household
expenditure) we find the primary income, but also the secondary income, 20, that
consists of unemployment benefits, 5, and other transfers (TRO = 15). The total
income of the household (Y) is equal to 420.
Finally, in the fourth column, (Expenditure), we find the household expenditure:
C1  64
C 2  150
Total taxes = 186 (income tax: 144, and consumption
taxes: 5 on commodity 1 and 37 on commodity 2) and SH  20 .
The ensuing equations will be given in the forthcoming document: Model.doc.
_____________
© Ecomod Network
13
6.
Actor 3: The Government
In figure 3 we summarize the decisions of the Government.
Figure 3. The decisions of the Government
Tariffs on
imports
Taxes on capital
Consumption
taxes
Taxes on labor
Income tax
Tax revenues
Unemployment
benefits
Other transfers
Other
government
expenditure
Government
savings
Transfers
CG1
CG2
KG
LG
In the fifth row of the SAM (Government) we find the revenues from tariffs (8 on the
imports of commodity 1 and 10 on those of 2), the taxes on capital and labor of
sector 1 (20) and those of sector 2 (46), and the taxes on consumption and on
income (186). Consequently, the total tax revenues are 270.
In the fifth column (Expenditure/government) we find the transfers to the household
(20), so that the other expenditure of the government is equal to 250.
It is assumed that the government maximizes a Cobb-Douglas utility function, with its
purchase of commodities ( CG1 and CG 2 ) and its demand for capital and labor (KG
and LG) as arguments, subject to the constraint that it spends 250.
In the fifth column (Expenditure/government) we find:
CG1  20
CG 2  50
KG  60
LG  120
The equations describing the behavior of the government will be given in Model.doc.
_____________
© Ecomod Network
14
7.
Actor 4: The Bank
In figure 4 we summarize the decisions of the bank.
Figure 4. The decisions of the Bank
SH
SG
SF
Savings
Demand for
investment
commodity 1
Demand for
investment
commodity 2
The household savings (SH), the government savings (SG) and the foreign savings
(SF) are allocated over the investment demand for commodities 1 and 2. To that end
the bank is assumed to maximize a Cobb-Douglas utility function with arguments I 1
and I 2 subject to the constraint that savings are equal to total investments.
In the sixth row of the SAM (Savings/investments) we find the household savings (SH
=20), whereas the government savings and foreign savings are equal to 0 (SG = SF
= 0). In the sixth column ( Expenditure/investment) we find the investments: I1  5
and I 2  15 .
The ensuing relations will be given in the forthcoming document: Model.doc.
_____________
© Ecomod Network
15
8.
Actor 5: The Rest of the World
In figure 5 we summarize the decisions of the Rest of the World.
Figure 5. The decisions of the Rest of the World
Exports of
commodity 1
Exports of
commodity 2
Export revenues
Foreign savings
Spending on
imports
Imports of
commodity 1
Imports of
commodity 2
The export revenues are found in the seventh column of the SAM (RoW) and are
equal to 10 and 36 respectively. Together with the foreign savings (SF=0) the total to
be spent on imports is 46. In the seventh row of the SAM (RoW) we find that the
imports of commodity 1 are equal to 16 and those of commodity 2 equal to 30.
The equations are to be found in the forthcoming document Model.doc.
_____________
© Ecomod Network
16
9.
The model
Variables:
K1 , K 2 , KS, PK, L1 , L 2 , LS, TS, UNEMP, PL, C1 , C2 , P1 , P2 , PCINDEX, X1 , X 2 , Y, CBUD, S, SF,
SG, SH, I1 , I 2 , CG1 , CG 2 , KG, LG, TAXR , TRF , TRO , PD1 , PD2 , XD1 , XD 2 , PDD1 , PDD 2 , XDD1 ,
XDD 2 , E1 , E 2 , M1 , M 2 , PM1 , PM 2 , PE1 , PE 2 , ER
Parameters:
Hi , HLES i , mps , Ii , Fi , Fi , aFi , io ij , G i , GK, GL,
A i , A i , aA i Ti , Ti , aTi , tc i , tk i , tl i , tm i , ty, trep, phillips
Household:
2
C i  H i  HLES i .[(1  tc i ).Pi ]1 .(CBUD   (1  tc j ).Pj .H j )
j1
SH  mps .(1  ty ).Y
LS  (TS  H 3 ) 
2


HLES 3
.[(1  ty ).PL]1 . CBUD   (1  tc j ).Pj .H j 
(1  HLES 3 )
j1


 PL1 / PCINDEX 1

 0
 1  phillips
0
 PL / PCINDEX

 UNEMP 1 / LS1

.
 1
0
0
 UNEMP / LS

2
PCINDEX 
 (1  tci ).Pi .Ci0
i 1
2
 (1  tc i0 ).Pi0 .Ci0
i 1
Investment demand:
S  SH  SG  ER.SF
Ii  Ii .Pi1.S
Firms:

K i  FiFi .[(1  tk i ).PK ]Fi . FiFi .[(1  tk i ).PK ]1Fi  (1  Fi )Fi .[(1  tl i ).PL]1Fi
.(XDi / aFi )
_____________
© Ecomod Network

Fi /(1 Fi )
.
17

Li  (1  Fi )Fi .[(1  tl i ).PL] Fi . FiFi .[(1  tk i ).PK ]1Fi  (1  Fi )Fi .[(1  tl i ).PL]1Fi

Fi /(1 Fi )
.(XDi / aFi )
Foreign sector:

XDD i  (1  Ai )A i .PDD i A i . AiA i .PM1i A i  (1  Ai )A i .PDD1i A i

M i  A iAi .PM iAi . A iAi .PM1iAi  (1  A i ) Ai .PDD1iAi

Ai /(1Ai )

XDD i  (1  Ti ) Ti .PDD iTi . TiTi .PE1iTi  (1  Ti ) Ti .PDD1iTi

E i  TiTi .PE iTi . TiTi .PE1iTi  (1  Ti ) Ti .PDD1iTi

Ti /(1Ti )


A i /(1 A i )
.(X i / aA i )
Ti /(1Ti )
.(XD i / aTi )
.(XD i / aTi )
PMi  1  tmi .ER.PWMZi
PE i  ER.PWEZ i
2
2
i 1
i 1
 PWMZ i .M i   PWEZ i .E i  SF
Government:
CG i  G i .Pi1 .(TAXR  TRF  SG)
KG  KG.PK 1 .(TAXR  TRF  SG)
LG  LG.PL1 .(TAXR  TRF  SG)
2
TAXR   tc i .Pi .C i  tk i .PK.K i  tl i .PL.L i  tm i .ER .PWMZ i .M i   ty.Y
i 1
TRF  trep.PL.UNEMP  PCINDEX.TRO
Market clearing:
K1  K2  KG  KS
* L1  L 2  LG  LS  UNEMP
X1  io11.XD1  io12.XD2  CG1  C1  I1
X2  io 21.XD1  io 22.XD2  CG2  C2  I2
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© Ecomod Network
.(Xi / aAi )
.
18
Income equations:
Y  PK.KS  PL.(LS  UNEMP )  TRF
CBUD  Y  ty.Y  SH
PD1.XD1  (1  tk1).PK.K1  (1  tl1).PL.L1  (P1.io11  P2.io 21).XD1
PD2.XD2  (1  tk 2 ).PK.K2  (1  tl 2 ).PL.L2  (P1.io 21  P2.io 22 ).XD2
Pi .Xi  PMi .Mi  PDDi .XDDi
PDi .XD i  PE i .E i  PDD i .XDD i
Comments:
The number of variables is equal to 49, whereas we dispose of 44 equations. In
order to arrive at the same number of variables and equations, we have to fix 5
variables. This is called: the closure of the model. In practice there are different
closure rules. In this example we have chosen to fix the following variables.
KS=170, TS=400, SF=0, SG=0, TRO=15
Walras’ law
After closing the model we dispose of 44 variables and equations. But it can be
shown that one of the equations is redundant, so that we dispose of 43 independent
equations. This is called the “Law of Walras”: when there are n markets, and n-1 of
them are cleared, then the n th market is cleared as well. In order to get a square
system we delete one of the equations. In the language of GAMS this is called
“commenting out”. We have chosen to comment out the market clearing of the labor
market. It is a standard procedure in GAMS to comment out another market clearing
equation than the labor market in order to check whether the same solution is found.
If not, an error has been made somewhere.
Consequently, we have to fix one more variable that is called the numeraire.
Usually, a price is fixed at one. We have chosen to fix:
PL = 1
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© Ecomod Network
19
10. An example of calibration
As example we use the calibration of the parameters of the demand relations for
capital and labor. In order to alleviate the notation, we suppress the index i pertaining
to the sector.
The cost to be minimized is equal to:
(1  tk ).PK.K  (1  tl).PL.L
(1)
whereas the original formulation of the linear-homogeneous CES production function
in capital and labor (K and L) reads:

XD  aF. F.K  F  (1  F).L F
1 / F
(2)
The first-order conditions for minimizing (1) subject to (2) are:
aF F .F.K (1 F) .XD (1 F)  1.(1  tk ).PK
(3)
and
aF F .(1  F).L(1 F) .XD (1 F)  1.(1  tl).PL
(4)
Dividing (3) by (4) yields, after rearrangement, the tangency condition:
F  K 
. 
1  F  L 
(1 F)

(1  tk ).PK
(1  tl).PL
(5)
In practice one usually uses the elasticity of substitution.
As is well-known, the Allen partial elasticity of substitution of the CES production
function is equal to:
F 
1
1  F
Then, (2) and (5) may be rewritten as:

XD  aF. F.K  (1 F) / F  (1  F).L (1 F) / F
and
F  K 
. 
1  F  L 
1 / F
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© Ecomod Network

(1  tk ).PK
(1  tl).PL
F /(1F)
(6)
(7)
20
In order to use the CES function we need to have values for F , F and aF .
Usually, the model builder has an idea about the value of the elasticity of substitution,
so that we have to calibrate the value of F and of aF .
From (7) we derive for the benchmark equilibrium:
1
F 
1
(8)
0  1 / F
K

.
(1  tk 0 ).PK 0  L0 
0
0
(1  tl ).PL
Having obtained the values of F and F , we use (6) to calibrate the value of aF :

aF  XD 0 / F.(K 0 )  (1 F) / F  (1  F).( L0 )  (1 F) / F
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© Ecomod Network
F /(1F)
(9)