SAM Multipliers

7/27/2019 SAM Multipliers

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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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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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( ) ( )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)


( ) ( )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
mailto:[email protected]:[email protected]:[email protected]:[email protected]

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SAM Multipliers

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