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INTERNATIONAL FOOD POLICY RESEARCH INSTITUTE
An Introduction with Exercises
Clemens BreisingerMarcelle ThomasJames Thurlow
SOCIAL ACCOUNTING MATRICES AND
MULTIPLIER ANALYSIS
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Overview
Exercise 1: Composition of a SAM
Exercise 2: Analysis of a SAM
Exercise 3: Input-output linkages and multiplier effects
Exercise 4: Unconstrained SAM multiplier analysis
Exercise 5: Constrained SAM multiplier analysis
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Exercise 1
Composition of a SAM
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Circular flow diagram of the economy
Government Productive
activities
Rest of world
Households Investment
Recurrentspending (G)
Consumptionspending (C)
Exports (E)
Imports (M)
Investmentdemand (I)
Direct taxes Fiscal surplus
Social transfers
Domestic private savings
Factor earnings(value-added)
Sales income
Remittances Foreign grantsand loans
Capital inflows
Indirect taxes
Factormarkets
Intermediatedemand
Commoditymarkets
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Basic structure of a SAM
Expenditure columns
Activities C1
Commods C2
Factors C3
HouseholdsC4
Government C5
Investment C6
Rest of world
C7 Total
I n c o m e r o w s
Activities R1
Domesticsupply
Activityincome
Commodities R2
Intermediatedemand
Consumption spending
(C)
Recurrentspending (G)
Investmentdemand (I)
Exportearnings (E)
Totaldemand
Factors R3
Value-addedTotal factor
income
Households R4
Factorpayments tohouseholds
Socialtransfers
Foreignremittances
Totalhousehold
income
Government R5
Sales taxesand import
tariffs
Directtaxes
Foreigngrants and
loans
Governmentincome
Savings R6
Privatesavings
Fiscalsurplus
Currentaccountbalance
Totalsavings
Rest of world R7
Importpayments
(M)
Foreignexchangeoutflow
Total Gross output Total supplyTotal factorspending
Totalhousehold
spending
Governmentexpenditure
Totalinvestment
spending
Foreignexchange
inflow
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Exercise 2
Analysis of a SAM
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2007 Ghana macro-SAM (millions of cedi)
Activities C1
Commods C2
Factors Households
C4 Govern.
C5 Investment
C6
Rest of
worldC7
Total Labor C3-1
Capital C3-2
Activities R1
24,996 24,996
Commodities R2
12,029 12,142 1,805 4,680 5,151 35,807
F a c t o r s
LaborR3-1 9,717 9,717
CapitalR3-2
3,250 3,250
Households R4
9,717 3,250 1,387 2,001 16,354
Government R5 2,372 940 739 4,052
Savings R6
3,272 860 548 4,680
Rest of world R7
8,439 8,439
Total 24,996 35,807 9,717 3,250 16,354 4,052 4,680 8,439
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Exercise 3
Economic linkages and multiplier effects
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Direct and indirect linkages
Directeffects
Indirecteffects
ConsumptionlinkagesExogenous
shock
Productionlinkages
Backwardlinkages
Forwardlinkages
Total impact of an exogenous demand shock = Direct effects + Indirect effects
Indirect effects = Production linkage effects + Consumption linkage effects
Consumption linkages: increased incomes generating consumptiondemand for other sectors products
Production linkages = Backward linkages + forward linkages
Backward linkages: additional demand generated by producers whenthey purchase intermediate inputs from other sectors
Forward linkages: supply of upstream producers with intermediate inputs
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Circular income flow in the multiplier process
Increase in
agriculturalexports
Increase inagriculturalproduction
Increase innonagricultural
production
Increase infactor
incomes andemployment
Increase inhousehold
incomes andconsumption
Direct effect
Productionlinkages
Consumptionlinkages
Indirect effects
Import leakage
Government
Rest of world
A
A B
C
Tax leakage
A: Output multipliers
B: GDP (value-added) multiplier
C: Income multiplier
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Exercise 4 Unconstrained SAM multiplier analysis
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Key assumptions for multipliers
Three assumptions behind multipliers 1. Prices are fixed and any changes in demand lead to
changes in physical output rather than prices.
2. Factor resources are unlimited or unconstrained , so thatany increase in demand is matched by increased supply.
3. Input coefficients of producers and consumption patterns
of households are unaffected by exogenous changes indemand (i.e., linkage effects are linear and there is nobehavioral change).
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SAM entries expressed as letters or symbols
Activities Commodities FactorsHouse-holds
Exogenousdemand Total
A1 A2 C1 C2 F H EA1 X1 X1 A2 X2 X2
C1 Z11 Z12 C1 E1 Z1 C2 Z21 Z22 C2 E2 Z2 F V1 V2 VH V1 + V2 YE L1 L2 S ETotal X1 X2 Z1 Z2 V Y E
We can replace the values appearing in the SAM with letters so that we can use them inderiving the multiplier formula
X = gross output of each activity (i.e., X 1 and X 2) Z = total demand for each commodity (i.e., Z 1 and Z 2) V = total factor income (equal to household income) Y = total household income (equal to total factor income)E = exogenous components of demand (i.e., government, investment and exports)
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Coefficient matrix (M-matrix)
Activities Commodities Factors House-holds
Exogenousdemand Total
A1 A2 C1 C2 F H E A1 b1= X1/Z1 X1 A2 b2= X2/Z2 X2
C1
a11 =Z11 /X1
a12=Z12 /X2
c1 = C1/Y
E1
Z1
C2 a21=Z21 /X1 a22=Z22 /X2 c2 = C2/Y E2 Z2 F v1=V1/X1 v2=V2/X2 V H 1 Y E l1 = L1/Z1 l2 = L2/Z2 s = S/Y E Total 1 1 1 1 1 1 E
We then divide each column through by its column total in order to derive a coefficientsmatrix (called M -matrix)
a = technical coefficients (i.e., input or intermediate shares in production)b = share of domestic output in total demandv = the share of value-added or factor income in gross outputl = share of the value of total demand from imports or commodity taxesc = household consumption expenditure sharess = household savings rate (i.e., savings as a share of total household income)
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Unconstrained multiplier formula (1)
Total demand Z in each sector is the sum of intermediate input demand, householdconsumption demand, and other exogenous sources of demand E
1 11 1 12 2 1 1Z =a X +a X +c Y+E
2 21 1 22 2 2 2Z =a X +a X +c Y+E
1 1 1X =b Z
From the SAM we know that gross output X is only part of total demand Z 2 2 2
X =b Z
We also know household income depends on the factors earnings shares in each sector
1 1 2 2 1 1 1 2 2 2Y=v X v X v b Z v b Z+ = +
We can now replace X and Y in the demand equations
( )1 11 1 1 12 2 2 1 1 1 1 2 2 2 1Z =a b Z +a b Z +c v b Z v b Z +E+( )2 21 1 1 22 2 2 2 1 1 1 2 2 2 2Z =a b Z +a b Z +c v b Z v b Z +E+
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Unconstrained multiplier formula (2)
From the previous slide
( )1 11 1 1 12 2 2 1 1 1 1 2 2 2 1Z =a b Z +a b Z +c v b Z v b Z +E+( )2 21 1 1 22 2 2 2 1 1 1 2 2 2 2Z =a b Z +a b Z +c v b Z v b Z +E+
Move all terms, except for exogenous demand E, onto the left- hand side
1 11 1 1 1 1 1 1 12 2 2 1 2 2 2 1Z -a b Z -c v b Z -a b Z -c v b Z =E
21 1 1 2 1 1 1 2 22 2 2 2 2 2 2 2-a b Z -c v b Z Z -a b Z -c v b Z E+ =
Finally, we group Z terms together
( ) ( )11 1 1 1 1 1 12 2 1 2 2 2 11-a b -c v b Z -a b -c v b Z =E+( ) ( )21 1 2 1 1 1 22 2 2 2 2 2 2-a b -c v b Z 1-a b -c v b Z E+ =
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Unconstrained multiplier formula (3)
From the previous slide
( ) ( )11 1 1 1 1 1 12 2 1 2 2 2 11-a b -c v b Z -a b -c v b Z =E+( ) ( )21 1 2 1 1 1 22 2 2 2 2 2 2-a b -c v b Z 1-a b -c v b Z E+ =
We can now use matrix algebra to convert the equations into matrix format
The first term is the identity matrix (I) minus the coefficient matrix (M). We can alsorename the other two vectors Z and E
( )I-M Z E=Finally, by rearranging terms, we arrive at the unconstrained multiplier formula.
( )-1Z I-M E=
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Exercise 5
Constrained SAM multiplier analysis
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Dropping a key assumption
Unconstrained multipliers assume that factor resources areunlimited or unconstrained, so that any increase indemand is matched by increased supply.
We now drop this assumption by preventing or constrainingchanges in some sectors production levels.
For the sectors with constrained supply, it is net exports thatnow decline when demand increases (i.e., imports increase
to replace any shortfall in domestic production).
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Constrained multiplier formula (1)
We derived the following demand equations in Exercise 4 ( ) ( )11 1 1 1 1 1 12 2 1 2 2 2 11-a b -c v b Z -a b -c v b Z =E+( ) ( )21 1 2 1 1 1 22 2 2 2 2 2 2-a b -c v b Z 1-a b -c v b Z E+ =
We now distinguish between sectors that can change their production level (Z 1), and thosesectors with supply constraints (Z 2).
Previously exogenous components of demand are now treated as exogenous (E 2) (i.e., netexports will now be able to change if domestic production cannot).
As with the unconstrained multiplier formula, we group exogenous components onto theright- hand side
( ) ( )11 1 1 1 1 1 1 12 2 1 2 2 21-a b -c v b Z =E a b c v b Z+ +( ) ( )21 1 2 1 1 1 2 22 2 2 2 2 2-a b -c v b Z E 1-a b -c v b Z- = -
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Constrained multiplier formula (2)
From the previous slide
( ) ( )11 1 1 1 1 1 1 12 2 1 2 2 21-a b -c v b Z =E a b c v b Z+ +( ) ( )21 1 2 1 1 1 2 22 2 2 2 2 2-a b -c v b Z E 1-a b -c v b Z- = -
This equation can be expressed in matrix format
The first term on the left-hand side is the identity matrix (I) minus an adjusted coefficientmatrix (M*). We will call the first term on the right-hand side the B-matrix.
Finally, by rearranging terms, we arrive at the constrained multiplier formula.
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This presentation accompanies the book:
SOCIAL ACCOUNTING MATRICESAND MULTIPLIER ANALYSIS
An Introduction with Exercises
Clemens Breisinger, Marcelle Thomas, and James Thurlow
Copyright 2009 International Food Policy Research Institute. All rightsreserved. For permission to republish, contact [email protected] .
www.ifpri.org
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