Introduction to Computable
General Equilibrium Analysis:
Input-Output Analysis Foundation
by
Adam Rose
CREATE and SPPD
University of Southern California
Modeling Needs
• Many economic phenomena and policy issues need
to be addressed at the macro level.
• Many of these are influenced by the interdependence
of the various sectors of the economy:
- economic development
- cost-push inflation
- cascading infrastructure failures
• We need models that are sectorally disaggregated &
sectorally linked thru both prices & quantities.
Key Questions
• What is an economic model?
• What are we modeling?
• What are the modeling alternatives?
• How do we choose the best model?
Economic Models
• A mathematical representation, based on economic
theory, of the workings of part or all of the economy
- micro, meso, macro
- simplification to focus on the essence of the
workings (not just “scale-model”; only selected
parts of the whole)
- an abstraction of reality
- used for analysis, prediction, policy evaluation
Alternative Modeling Approaches
• Econometric
- based on solid data; but data intensity an obstacle
- does not explicitly model interactions
• Input-Output
- prevalent non-survey (data reduction) models
- limitations: linear, no behavior, no mkts/prices
• Computable General Equilibrium
- maintains I-O strengths; overcomes limitations
- data base not as solid as econometric (calibration)
Economic Model Choice
• Strategic elements in model selection:
- policy question (applicability)
- relevant assumptions (behavior, spatial
resolution, role of markets, constraints)
- data availability
- other criteria (cost, transparency, etc.)
Evaluative Criteria
•
•
•
•
•
•
•
•
Accuracy
Scope
Detail
Transparency
Manageability
Flexibility
Cost
Other
Evaluating Alternative Models
Criteria
I-O
CGE
ME
Accuracy
Fair
Good
Good
Cost
Excellent
Fair
Fair
Scope
Fair
Good
Good
Detail
Excellent
Excellent
Good
Flexibility
Fair
Excellent
Good
Transparency
Excellent
Good
Fair
Manageability
Good/Excellent
Fair/Good
Good
Understanding CGE Models
• Theoretical Foundation: Walrasian GE
• Empirical Foundation:
- I-O accounts for production inputs
- Social Accounting Matrix for hh & institutions
- Data transfer for elasticities
• Solution Algorithms
- non-linear programming
- variant of fixed-point theorem
Overview of CGE
• State-of-the-art impact analysis method
• Relative advantages
- workings of markets & prices
- behavior of individual decision-makers
- substitution & other non-linearities
- ability to accommodate engineering data
• Some disadvantages being overcome
Key Underpinnings of CGE
• Input-Output Analysis
• Social Accounting Matrices
• Mathematical Programming
• General Equilibrium Theory
Definition of Input-Output Analysis
Basic Model: A static, linear model of all purchases
and sales between sectors of an economy, based
on the technological relationships of production.
Ultimate Version: A dynamic, non-linear model of
all purchases and sales, both market and nonmarket, between sectors of economies, based on
the technological relationships of production and
other variables that can be quantified.
Input-Output Analysis--Rich History
• Worthy of Nobel Prize to Wassily Leontief
• Wealth of empirical data
• Still major tool of impact analysis
• Many superior applications
Advantages of Input-Output Models
• Organizational framework for data
• Comprehensive accounting of all inputs
• Displays economic structure
• Reveals economic linkages
• Calculates total (direct, indirect, & induced impacts)
• Computational ease
• Readily extended (institutions, pollution, etc.)
• Can accommodate engineering data
• Empirical models readily available
Disadvantages of I-O Models
• Prices play a secondary role
• Lack behavioral content
• Linearities are difficult to overcome
• Lacks forecasting ability
Three Versions of the Basic I-O Table
1. Transactions Table (annual physical or dollar
flows)
2. Structural Matrix (direct input requirements
per unit of output)
3. Leontief Inverse Matrix (total input
requirements per unit of output)
Assumptions Underlying
the Basic I-O Model
1. One-to-one correspondence
A. Uniqueness of production
B. No joint-products
2. Proportionality of inputs and outputs
3. No externalities
Mathematical Presentation
Basic Balance Identity:
n
X i   X ij  Yi
i 1 . . . n
j1
where
X i = total gross output of industry i
Yi = final demand from industry i
X ij = amount of good i used in the
production of good j
(1a)
Math Presentation (continued)
The Structural Model:
a ij 
let
X ij
Xj
where
a ij = the direct input requirement of good i
needed to produce one unit of good j
then
n
X i   a ijX j  Yi
i 1 . . . n
j1
which is short-hand notation for:
X1 = a11 X1  a12 X2 . . . a1n Xn  Y1
X 2 = a 21 X1  a 22 X2 . . . a 2n Xn  Y2
Xn =
a n1 X1  a n2 X2 . . . a nn Xn  Yn
(1b)
Elements of the Leontief Inverse
B   bij 
bij = the total amount of good i required directly and indirectly to provide
one unit of good j for final demand
example: if b rs = .38 (where r represents iron ore and s represents steel)
.24 = direct input requirements (iron ore in steel production)
.0 = first round indirect (no iron ore needed to produce iron ore)
.01 = second round indirect (e.g., iron ore needed to produce steel
to mine iron ore)
.06 = third round indirect (e.g., iron ore needed to produce steel
needed to produce mining equipment to mine iron ore)
.02 = fourth round indirect (e.g., iron ore needed to produce steel
needed to produce locomotives to transport mining equipment
to iron ore mines)
Matrix Presentation of I-O
Basic Balance Identity:
X  AX  Y
(2a)
Basic I-O Problem:
X  AX  Y
(2b)
Solution:
I  A X  Y
I  A I  A X  I  A
1
(3)
1
Y
(4)
IX  I  A Y
(5)
X  I  A Y
(6)
1
1
where
I  A
1
 B   bij 
1
 I  A  A 2  A3  A 4 . . .
also
I  A
Open vs. Closed I-O Models
Open I-O Model--Consists of only the intermediate sectors in the structural matrix (direct
requirements coefficients) and the Leontief inverse (total requirements coefficients).
Therefore, only capable of estimating indirect (interindustry) effects of an exogenous
stimulus:
X = AX + Y
Closed I-O Model--Consists of the intermediate sectors plus one of more of the normally
exogenous final demand sectors (typically consumption) plus the corresponding payments
sector (typically household income). Therefore, capable of estimating indirect
(interindustry) and induced (typically income/spending) effects of an exogenous stimulus:
I  A * H 
C


*H
1
h


R
 Y* 
X 
 


X n1

Y*
 n1 
Basic Multiplier
Y=C+I
I = Ia
C = a + bY
1
1
1
M


1  mpc 1  b leakages
e.g., when b = .75, M = 4
X = (I-A)-1Y
∆X = (I-A)-1∆Y
∆X = M • ∆Y
Input-Output Multipliers
Basic Concept: Total Impacts (throughout the economy)
Direct Impacts (in one sector)
Two Formats:
• Partial derivative (numerator & denominator in same units)
e.g.,
Total Employment Change
Direct Employment Change
• Standardized (all impacts expressed in terms of ∆X)
e.g.,
Total Employment Change
Per Unit Change in Output
Multiplier Types
Type I:
Total impacts include direct & indirect effects
(computed with the open I-O Table)
Type II:
Total impacts include direct, indirect & induced
(computed with the closed I-O Table)
Type III:
Total impacts include direct, indirect & induced
(computed with closed Table & marginal consumption
coefficients rather than average coefficients)
Type X:
Closed to other elements of Final Demand
(e.g., closed w/ respect to investment: dynamic multiplier)
Type SAM: Total impacts with interaction among institutions
(computed with Social Accounting Matrix)
I-O Multiplier Calculations
Hypothetical I-O Table
1
2
HH
OFD
X
1
20
45
30
5
100
2
40
15
30
65
150
HH
20
60
10
10
100
OVA
20
30
30
--
80
X
100
150
100
80
430
(I-A)-1
A
.2
.3
.4
.1
.2
.4
1.5
.67
(1-A*)-1
A*
.50
.2
.3
.3
2.09
1.18
1.09
1.33
.4
.1
.3
1.27
2.00
1.09
.2
.4
.1
1.03
1.15
1.82
n
Type I Output Mutiplier 
b
ij
i1
n
Type II Output Multiplier 
b
i1
*
ij
Income Multipliers
Direct Income Effect
 ahi
.2  ahi
Direct & Indirect Effect
n
  ahibij
2
.2(1.5)  .4(.67)  .568   ahibi1
i
i
Type I Income Multiplier
n
ahibij
i
ahi

.568  .2  2.84 
a
Direct & Indirect & Induced Effects
 bhj*
1.03  bh*1
Type II Income Multiplier

bh*
j
ah1
bh*1
1.03  .2  5.15 
ahl
b
hi i1
ah1
Definitions & Conventions of
Input-Output Tables
1. Valuation of transactions in producer prices:
purchasers P = producer P + transport C
+ trade margin
2. Trade and transport margins:
cost of doing business only
3. Secondary products:
several conventions
Definitions & Conventions (cont.)
4. Dummy industries
constant mix of small items
5. Inventories
in terms of industries producing them
6. Trade
several conventions, but main one:
competitive (comparable, transferred)
non-competitive (non-comparable, directly
allocated)
I-O Table Construction
1.
Select a time period (usually 1 year)
2.
Classify major components
a.
b.
c.
d.
Industry categories
Final demand categories
Income payment categories
Trade categories (imports and exports)
3.
Establish sectoral control totals
4.
Tabulate intersectoral flows
a. Production requirements
b. sales distributions
5.
Cross-check and reconcile data
Regional I-O Models
Why a separate category?
• Superficial answer: sub-national unit
• More accurate answer: Open economy vs. closed economy
Affects the choice of structural coefficients:
• Technical coefficients total direct requirements
(national, irrespective of geographic origin)
• Trade coefficients—only counts goods produced &
(intraregional, used within the region)
Columns from Hypothetical Nationa
& Regional I-O Tables
coal
iron
water
labor
imports
(1)
national technology
(2)
regional actual technology
(3)
actual regional + trade
a nij
arij
rij
mij
0 (0)
10 (.l)
10 (.l)
40 (.4)
40 (.4)
100
30 (.3)
10 (.l)
0
0
40
200
200

100
500
1,000
(.2)
(.2)
(.1)
(.5)
-

30
20
10
40
100
(.3)
(.2)
(.l)
(.4)
-
(2')
approximate
regional technology
(3')
approximate
regional + trade
arij '
r'ij

20
20
10
50
100
(.2)
(.2)
(.l)
(.5)
-
0
10
10
50
30
100
(0)
(.1)
(.1)
(.5)
(.3)
mij
20 (.2)
10 (.1)
0
0
0
30
Major Regional Interregional &
Multiregional Input-Output Models
1.
Pure Regional Model
regional specific technology and input requirements
(survey based tables--Isard; Miernyk; Bourque)
R
2.
Pure Regionalized Model
national technology as basis for regionalized input requirements
(non-survey tables--Schaffer and Chu)
AR or PA
3.
Pure Interregional Model
regional specific origins and destinations
(theoretical ideal--Isard)
AML
4.
Multiregional Model
national technical coefficients, but regional mix
(HRIO--Chanery; Moses; Polenske)
AL & CLM
5.
Balanced (Multi-) Regional Model
supply-demand balance in regional and national markets
(Leontief)
ARN & PRXN
Supply-Demand Pool Technique for
Generating Regional I-O Models
1.
Scale down national flows to conform to regional control totals (multiply input flows in
each column by ratio of regional to national gross output).
2.
Sum each row of the scaled down flow table to determine total regional demand for each
sector's output.
3.
Subtract the total demand for each sector's output from its corresponding sectoral gross
output total (regional supply).
a. If excess demand for a sector's output is negative, there is an exportable surplus and no
further adjustment is needed in that sector's (row) flows. The exportable surplus, is
entered as the sector's row entry in a single "Export Column." (Also, this sector's output
will therefore not be imported.)
b. If excess demand for a sector's output is positive, there is an import deficit. Apportion
imports proportionally across all buyers (columns).
i. Multiply each row (including final demand elements) by the ratio of its total sector
supply and total sector demand. The result is a row vector of intraregional
tradeflows for that sector's output.
ii. Subtract the result of the prior calculation from the scaled down row entries
Note that the methodology invokes the "no cross-hauling" assumption--no sector's output
will be both exported and imported.
4.