Emerta A. Aragie; November 2023; Addis Ababa, Ethiopia; Introduction to Multiplier Analysis

1. Introduction to Multiplier Analysis
Emerta A. Aragie
November 2023
Addis Ababa, Ethiopia

2. • Outline:
• Economic linkages and multiplier effects
• Unconstrained SAM multiplier analysis

3. Economic linkages and multiplier effects
• Economic linkages are fairly static and are determined by the structural
characteristics of an economy
• production technologies and the composition of households’ consumption
• Multiplier effects capture the combined effects of economic linkages over
a period of time derived from an exogenous change.
• Multiplier effect refers to the idea that an initial change in demand can
cause a further change in output.
• In building a SAM multiplier model, the first step is to decide which
accounts should be exogenous or endogenous.
• It is customary to consider the government, export and investment
demand as exogenous
• Largely policy determined (?gov?) or outside the domestic economy.
• Hence “exogenous demand-side shocks” for the multiplier analysis can
refer to changes in export demand, government spending, or investment
demand.

4. • Total multiplier effect a shock = Direct effects + Indirect effects
• Direct effects are those effects on the sector directly affected by the
shock.
• Eg. An exogenous increase in manuf. would increase production, supply and VA to
manuf.
• Indirect effects = Consumption linkage + Production linkage effects
• Consumption linkages (dd side): when increased incomes generate
consumption demand for products of other sectors.
• Production linkages (ss side)= Backward linkages + forward linkages
• Backward linkages: the expanding sector creates additional demand for intermediate
inputs from other sectors
• E.g. processing and agric sector; metallic and non-metallic manufacturing.
• Forward linkages: supply of cheap intermediate inputs to downstream sectors.

5. • A figurative representation:
• Total multiplier effects are the sum of direct and indirect effects.
Direct
effects
Indirect
effects
Consumption
linkages
Exogenous
shock
Production
linkages
Backward
linkages
Forward
linkages

6. Increase in
agricultural
production
Increase in
nonagricultural
production
Increase in
factor
incomes &
employment
Increase in
household
incomes and
consumption
Direct effect
Production
linkages
Consumption
linkages
Indirect effects
Import leakage
Government
Rest of world
A
A B
C
Tax leakage
Three types:
A: Output multipliers
B: GDP (value-added) multiplier
C: Income multiplier
Increase in
agricultural
exports
Shock

7. • A number of structural characteristics of an economy determine the size
of a multiplier:
• The size of forward and backward production linkages
• If sectors are interdependent, multipliers become higher.
• The share of imported commodities in households’ consumption
• Imports are leakages from the circular flow
• Government taxes on factor incomes
• Reduces the consumption linkage (one component of multipliers)

8. Calculating Linkage Effect → Exercise 3 (Task 3)
Complete “Task 3 Worksheet.xls” in the
“Exercises” folder. The solution file is in the
“Solutions” sub-folder.
Task 3:
Calculating Round-by-Round Linkage
Effects
• Activity: Calculate backward production linkage effects during each round of
the circular flow of income.
• As a sector grows, it creates demand for products of other sectors.
• determine how downstream sectors benefit when agric. production increases (and
demands more inputs)
• what causes the multiplier effect?
• The importance of the input coefficient

9. Unconstrained SAM Multiplier Analysis
• We have three key assumptions for the multipliers:
• Prices are fixed and any changes in demand lead to changes in physical output rather
than prices.
• Alternative to these are fixed quantity multipliers – price models since 1995 by Roland-
Hurst and co.
• Factor resources are unlimited or unconstrained, so that any increase in demand is
matched by increased supply.
• Input coefficients (I-O coefficients) of producers and consumption patterns of
households are unaffected by exogenous changes in demand
• I.e., linkage effects are linear and there is no behavioral change.
• No changes in relative prices

10. • Assume the following 7X7 aggregate SAM.
Activities Commodities Factors House-
holds
Exogenous
demand
Total
A1 A2 C1 C2 F H E
A1 𝑋1 𝑋1
A2 𝑋2 𝑋2
C1 𝑍11 𝑍12 𝐶1 𝐸1 𝑍1
C2 𝑍21 𝑍22 𝐶2 𝐸2 𝑍2
F 𝑉1 𝑉2 V
H V Y
E 𝐿1 𝐿2 S E
Total 𝑋1 𝑋2 𝑍1 𝑍2 V Y E

11. • We divide each column through by its column total in order to derive a
coefficients matrix (called “M-matrix”)…
• a = technical coefficients (i.e., input or intermediate shares in production)
• b = share of domestic output in total supply
• v = the share of value-added or factor income in gross output
• l = share of the value of total supply from imports or commodity taxes
• c = household consumption expenditure shares
• s = household savings rates
Activities Commodities Factors House-
holds
Exogenous
demand
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 = 𝐸1/E 𝑍1
C2 𝑎21 = 𝑍21/𝑋1 𝑎22 = 𝑍22/𝑋2 𝑐2 = 𝐶2/Y 𝑒2 = 𝐸2/E 𝑍2
F 𝑣1 = 𝑉1/𝑋1 𝑣1 = 𝑉2/𝑋2 V
H 1 Y
E 𝐼1 = 𝐿1/𝑍1 𝐼2 = 𝐿2/𝑍2 s = S/Y E
Total 1 1 1 1 1 1 1

12. Unconstrained multiplier formula (1)
• Total demand Z in each sector is the sum of intermediate input demand,
household consumption demand, and other exogenous sources of demand
E…
• From the SAM we know that gross output X is only part of total demand Z…
• We also know household income depends on the factors earnings shares in
each sector…
1 11 1 12 2 1 1
Z =a X +a X +c Y+E
2 21 1 22 2 2 2
Z =a X +a X +c Y+E
1 1 1
X =b Z 2 2 2
X =b Z
1 1 2 2 1 1 1 2 2 2
Y=v X v X v b Z v b Z
+ = +

13. Unconstrained multiplier formula (2)
• We can now replace X and Y in the demand equations…
• Move all terms, except for exogenous demand E, onto the left-hand side…
• Finally, we group Z terms together…
( )
1 11 1 1 12 2 2 1 1 1 1 2 2 2 1
Z =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 2
Z =a b Z +a b Z +c v b Z v b Z +E
+
1 11 1 1 1 1 1 1 12 2 2 1 2 2 2 1
Z -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
+ =
( ) ( )
11 1 1 1 1 1 12 2 1 2 2 2 1
1-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
+ =

14. • 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 also rename the other two vectors Z and E…
• Finally, by rearranging terms, we arrive at the unconstrained multiplier
formula.…
• We will implement these formulas in Excel in building the multiplier model.
( )
I-M Z E
=
( )-1
Z I-M E
=

15. Unconstrained multiplier model → Exercise 4 (Task 4)
Complete “Task 4 Worksheet.xls” in the
“Exercises” folder. The solution file is in the
“Solutions” sub-folder.
Task 4:
Constructing an Unconstrained
Multiplier Model
• Activity: Calculate unconstrained multipliers
• Translate the mathematical equations presented above into Excel
• Use the MINVERSE and MMULT Excel functions
• Answer the questions in red at the bottom of the worksheet
• Look at the key check info in blue to guide your computation.
• Interpret the multipliers