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Task 4: Constructing an unconstrained multiplier model
2007 SAM for Ghana with 3 sectors
Units: Millions of new Ghanaian Cedis (current prices)aagr aind asrv cagr cind csrv lab cap
aagr 7,070
aind 9,084
asrv 8,842
cagr 606 831 126
cind 599 3,423 1,889
csrv 1,313 880 2,361
lab 4,215 2,254 3,248
cap 336 1,696 1,218
hrur 5,054 1,248
hurb 4,663 2,002
gov 160 1,869 343
s-i
row 575 7,450 415
tot 7,070 9,084 8,842 7,805 18,403 9,599 9,717 3,250
PART 1: Calculating the inverted coefficient matrix
Coefficient matrix (M)
Calculate coefficients by dividing each column through by its column total (see Table 4).
aagr aind asrv cagr cind csrv lab cap
aagr 0.91
aind 0.49
asrv 0.92
cagr 0.09 0.09 0.01
cind 0.08 0.38 0.21
csrv 0.19 0.10 0.27
lab 0.60 0.25 0.37
cap 0.05 0.19 0.14
hrur 0.52 0.38hurb 0.48 0.62
gov 0.02 0.10 0.04
s-i
row 0.07 0.40 0.04
tot 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
Indentity matrix (I)
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Create an identity matrix by placing the value 1 in the diagonal cell entries
aagr aind asrv cagr cind csrv lab cap
aagr 1.00
aind 1.00
asrv 1.00
cagr 1.00
cind 1.00csrv 1.00
lab 1.00
cap 1.00
hrur
hurb
gov
s-i
row
Intermediate step (I-M)
Subtract the matrix from the identity matrix
aagr aind asrv cagr cind csrv lab cap
aagr 1.00 -0.91
aind 1.00 -0.49
asrv 1.00 -0.92
cagr -0.09 -0.09 -0.01 1.00
cind -0.08 -0.38 -0.21 1.00
csrv -0.19 -0.10 -0.27 1.00
lab -0.60 -0.25 -0.37 1.00
cap -0.05 -0.19 -0.14 1.00hrur -0.52 -0.38
hurb -0.48 -0.62
gov -0.02 -0.10 -0.04
s-i
row -0.07 -0.40 -0.04
-0.04
Inverted coefficient matrix (I-M)-1
Invert the above matrix using the Excel "MINVERSE" formula.
aagr aind asrv cagr cind csrv lab cap
aagr 1.54 0.47 0.43 1.39 0.23 0.39 0.51 0.48
aind 0.49 1.58 0.53 0.44 0.78 0.49 0.46 0.45
asrv 0.64 0.49 1.70 0.58 0.24 1.56 0.44 0.44
cagr 0.59 0.52 0.47 1.54 0.25 0.43 0.56 0.53
cind 0.99 1.18 1.08 0.90 1.58 1.00 0.93 0.92
csrv 0.70 0.53 0.76 0.63 0.26 1.70 0.48 0.48
lab 1.27 0.85 1.01 1.15 0.42 0.93 1.58 0.56
cap 0.25 0.38 0.35 0.23 0.19 0.33 0.17 1.17
hrur 0.76 0.59 0.66 0.69 0.29 0.61 0.88 0.74
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hurb 0.77 0.64 0.70 0.69 0.32 0.65 0.86 0.99
gov 0.22 0.22 0.22 0.22 0.21 0.24 0.21 0.22
s-i 0.31 0.25 0.27 0.28 0.12 0.25 0.35 0.35
row 0.48 0.54 0.51 0.50 0.67 0.51 0.44 0.43
PART 2: The exgenous shock vector
Exogenous demand shock matrix (E)
cagr cind csrv Hint
aagr This matrix shows the size of the exogenous d
aind It is initially set equal to one, which means a 1
asrv You can change the size of the shock to any n
cagr 1.00
cind 1.00 Remember the unit of the shock and hence th
csrv 1.00 (i.e., millions of Ghanaian Cedi at 2007 prices)
lab
cap It is necessary for matrix calculations that all
hrur Note though that we are hiding all zero entrie
hurb
gov
s-i
row
PART 3: The multiplier effects
Multiplier effects Z = (I-M)-1 E
Use the Excel "MMULT" formula to calculate the multiplier effects
cagr cind csrv
aagr 1.39 0.23 0.39 Output multipliers
aind 0.44 0.78 0.49
asrv 0.58 0.24 1.56
cagr 1.54 0.25 0.43
cind 0.90 1.58 1.00
csrv 0.63 0.26 1.70
lab 1.15 0.42 0.93 GDP or value-added multipliers
cap 0.23 0.19 0.33hrur 0.69 0.29 0.61 Income multipliers
hurb 0.69 0.32 0.65
gov 0.22 0.21 0.24
s-i 0.28 0.12 0.25 0.25
row 0.50 0.67 0.51
Total multiplier effects by type of multiplier
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Output 2.42 1.25 2.45
GDP 1.38 0.61 1.26 Note that income and GDP multipliers should
Income 1.38 0.61 1.26 factor taxes in the Ghana SAM (i.e., all value-a
0.61
Q1: Which sector has the largest output multiplier?
A1: Services
Q2: Which sector has the largest income multiplier?
A2: Agriculture
Q3: What do you think is the main reason why the multipliers for industry so much lower than f
A3: There are large import-leakages for the industry, since so much of industrial demand is supp
Q4: What is the output multiplier effect of a by 35 million cedi increase in agricultural export de
A4: 84.62 million cedi
(to answer this question you need to replace the "1" in the agricultural column of the exoge
with a "35" - you can then read the total output multiplier from the table above).
Q5: Which sector generates the largest income multipler effects for urban households?
A5: Services
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hrur hurb gov s-i row tot SAM legend
7,070 aagr Activity - a
9,084 aind Activity - in
8,842 asrv Activity - se
2,453 1,773 2,016 7,805 cagr Commodit
2,553 3,238 4,680 2,020 18,403 cind Commodit
705 1,421 1,805 1,115 9,599 csrv Commodit
9,717 lab Factor - lab
3,250 cap Factor - ca
547 409 7,257 hrur Household
840 1,592 9,097 hurb Household
940 739 4,052 gov Governme
1,547 1,725 860 548 4,680 s-i Saving/inve
8,439 row Rest of the
7,257 9,097 4,052 4,680 8,439 tot Column an
hrur hurb gov s-i row Hint
The final three columns in the coe
This means that we must leave th
round-by-round indirect linkage ef
0.34 0.19
0.35 0.36
0.10 0.16
0.10
0.21 0.19
1.00 1.00
1.00 0.00
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hrur hurb gov s-i row
1.00
1.00
1.00
1.00
1.00
1.00
hrur hurb gov s-i row Hint
You must use Excel matrix algebra
To do this, do the following four st
-0.34 -0.19 1. Highlight the block where you w
-0.35 -0.36 2. Then types equals (=) followed
-0.10 -0.16 3. Then subtract (-) the coordinate
4. Finally, hold down the control (
1.00 Now if you move your cursor over
1.00 has "curly" parentheses or bracket
-0.10 1.00 a formula is matrix algebra.
-0.21 -0.19 1.00
1.00
1.00
hrur hurb gov s-i row
0.59 0.41
0.47 0.44
0.43 0.44
0.65 0.46
0.96 0.89
0.47 0.48
0.63 0.52
0.18 0.16
1.39 0.33
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0.41 1.35
0.17 0.26 1.00
0.37 0.33 1.00
0.46 0.42 1.00
0.42 1.00
emand shock for each commodity.
million cedi increase in demand for each commodity.
mber.
e multiplier is in value terms
.
f the cell entries have a value, even if is zero.
s (see the Tools-Options drop-down menu).
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be the same since there are no
dded is paid to households).
r other sectors?
lied by imported goods
and?
nous shock matrix
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riculture
dustry
rvices
- agriculture
- industry
- services
or
ital
- rural
- urban
t
stment
world
row total
ficient matrix are exogenous accounts in our model.
se accounts as zero, since they do not generate
fects. All other columns should sum to one.
7/30/2019 Task 4 Solution
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commands for this exercise.
eps when entering the formula:
ant your answer to be (B67:N79).
y the coordinates for the entire identity matrix (B49:N61).
s for the entire coefficient matrix (B30:N42) (leave out the total row)
TRL) and SHIFT keys, and then press enter
the matrix to the left you should see that your formula
s around it (i.e., { and }). This is how Excel shows that
7/30/2019 Task 4 Solution
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Task 4: Constructing an unconstrained multiplier model
2007 SAM for Ghana with 3 sectors
Units: Millions of new Ghanaian Cedis (current prices)aagr aind asrv cagr cind csrv lab cap
aagr 7,070
aind 9,084
asrv 8,842
cagr 606 831 126
cind 599 3,423 1,889
csrv 1,313 880 2,361
lab 4,215 2,254 3,248
cap 336 1,696 1,218
hrur 5,054 1,248
hurb 4,663 2,002
gov 160 1,869 343
s-i
row 575 7,450 415
tot 7,070 9,084 8,842 7,805 18,403 9,599 9,717 3,250
PART 1: Calculating the inverted coefficient matrix
Coefficient matrix (M)
Calculate coefficients by dividing each column through by its column total (see Figure 5).
aagr aind asrv cagr cind csrv lab cap
aagr 0.91
aind 0.49
asrv 0.92
cagr 0.09 0.09 0.01
cind 0.08 0.38 0.21
csrv 0.19 0.10 0.27
lab 0.60 0.25 0.37
cap 0.05 0.19 0.14
hrur 0.52 0.38hurb 0.48 0.62
gov 0.02 0.10 0.04
s-i
row 0.07 0.40 0.04
tot 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
Indentity matrix (I)
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Create an identity matrix by placing the value 1 in the diagonal cell entries
aagr aind asrv cagr cind csrv lab cap
aagr 1.00
aind 1.00
asrv 1.00
cagr 1.00
cind 1.00csrv 1.00
lab 1.00
cap 1.00
hrur
hurb
gov
s-i
row
Intermediate step (I-M)
Subtract the matrix from the identity matrix
aagr aind asrv cagr cind csrv lab cap
aagr 1.00 -0.91
aind 1.00 -0.49
asrv 1.00 -0.92
cagr -0.09 -0.09 -0.01 1.00
cind -0.08 -0.38 -0.21 1.00
csrv -0.19 -0.10 -0.27 1.00
lab -0.60 -0.25 -0.37 1.00
cap -0.05 -0.19 -0.14 1.00hrur -0.52 -0.38
hurb -0.48 -0.62
gov -0.02 -0.10 -0.04
s-i
row -0.07 -0.40 -0.04
Inverted coefficient matrix (I-M)-1
Invert the above matrix using the Excel "MINVERSE" formula.
aagr aind asrv cagr cind csrv lab cap
aagr 1.54 0.47 0.43 1.39 0.23 0.39 0.51 0.48
aind 0.49 1.58 0.53 0.44 0.78 0.49 0.46 0.45
asrv 0.64 0.49 1.70 0.58 0.24 1.56 0.44 0.44
cagr 0.59 0.52 0.47 1.54 0.25 0.43 0.56 0.53
cind 0.99 1.18 1.08 0.90 1.58 1.00 0.93 0.92
csrv 0.70 0.53 0.76 0.63 0.26 1.70 0.48 0.48
lab 1.27 0.85 1.01 1.15 0.42 0.93 1.58 0.56
cap 0.25 0.38 0.35 0.23 0.19 0.33 0.17 1.17
hrur 0.76 0.59 0.66 0.69 0.29 0.61 0.88 0.74
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hurb 0.77 0.64 0.70 0.69 0.32 0.65 0.86 0.99
gov 0.22 0.22 0.22 0.22 0.21 0.24 0.21 0.22
s-i 0.31 0.25 0.27 0.28 0.12 0.25 0.35 0.35
row 0.48 0.54 0.51 0.50 0.67 0.51 0.44 0.43
PART 2: The exgenous shock vector
Exogenous demand shock matrix (E)
cagr cind csrv Hint
aagr This matrix shows the size of the exogenous d
aind It is initially set equal to one, which means a 1
asrv You can change the size of the shock to any n
cagr 1.00
cind 1.00 Remember the unit of the shock and hence th
csrv 1.00 (i.e., millions of Ghanaian Cedi at 2007 prices)
lab
cap It is necessary for matrix calculations that all
hrur
hurb
gov
s-i
row
PART 3: The multiplier effects
Multiplier effects Z = (I-M)-1 E
Use the Excel "MMULT" formula to calculate the multiplier effects
cagr cind csrv
aagr 1.39 0.23 0.39 Output multipliers
aind 0.44 0.78 0.49
asrv 0.58 0.24 1.56
cagr 1.54 0.25 0.43
cind 0.90 1.58 1.00
csrv 0.63 0.26 1.70
lab 1.15 0.42 0.93 GDP or value-added multipliers
cap 0.23 0.19 0.33hrur 0.69 0.29 0.61 Income multipliers
hurb 0.69 0.32 0.65
gov 0.22 0.21 0.24
s-i 0.28 0.12 0.25
row 0.50 0.67 0.51
Total multiplier effects by type of multiplier
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Output 2.42 1.25 2.45
GDP 1.38 0.61 1.26 Note that income and GDP multipliers should
Income 1.38 0.61 1.26 factor taxes in the Ghana SAM (i.e., all value-a
Q1: If we make government consumption demand endogenous in the multiplier model, what ha
A1: Increase
Q2: Why do the multipliers change when govenrment is made endogenous?
A2: Before when tax revenues rose during each round of the multiplier process, we did not let g
Now that government spending is endogenous, any increase in taxes raises government spe
for services, which now increase production more than before. The multiplier effects are th
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hrur hurb gov s-i row tot SAM legend
7,070 aagr Activity - a
9,084 aind Activity - in
8,842 asrv Activity - se
2,453 1,773 2,016 7,805 cagr Commodit
2,553 3,238 4,680 2,020 18,403 cind Commodit
705 1,421 1,805 1,115 9,599 csrv Commodit
9,717 lab Factor - lab
3,250 cap Factor - ca
547 409 7,257 hrur Household
840 1,592 9,097 hurb Household
940 739 4,052 gov Governme
1,547 1,725 860 548 4,680 s-i Saving/inve
8,439 row Rest of the
7,257 9,097 4,052 4,680 8,439 tot Column an
hrur hurb gov s-i row Hint
The final three columns in the coe
This means that we must leave th
round-by-round indirect linkage ef
0.34 0.19
0.35 0.36
0.10 0.16
0.10
0.21 0.19
1.00 1.00
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hrur hurb gov s-i row
1.00
1.00
1.00
1.00
1.00
hrur hurb gov s-i row Hint
You must use Excel matrix algebra
To do this, do the following four st
-0.34 -0.19 1. Highlight the block where you w
-0.35 -0.36 2. Then types equals (=) followed
-0.10 -0.16 3. Then subtract (-) the coordinate
4. Finally, hold down the control (
1.00 Now if you move your cursor over
1.00 has "curly" parentheses or bracket
-0.10 1.00 a formula is matrix algebra.
-0.21 -0.19 1.00
1.00
hrur hurb gov s-i row
0.59 0.41
0.47 0.44
0.43 0.44
0.65 0.46
0.96 0.89
0.47 0.48
0.63 0.52
0.18 0.16
1.39 0.33
7/30/2019 Task 4 Solution
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0.41 1.35
0.17 0.26 1.00
0.37 0.33 1.00
0.46 0.42 1.00
emand shock for each commodity.
million cedi increase in demand for each commodity.
mber.
e multiplier is in value terms
.
f the cell entries have a value, even if is zero.
PART 4: Deciding on exogenous demand components
Exogenous demand components
Enter a one if the component of demand is exogenous in the model (zero otherwise).
gov s-i row
1 1 1
This extended setup allows you to choose exogenous demand components.
If 1 is chosen, then the respective component is treated as exogenous.
If 0 is chosen, then the respective component is treated as endogenous.
Please see code in the M matrix' exogenous accounts above to see how we implement this.
You cannot set all demand components to be endogenous at the same time.
By default we set government equal to zero, so that it is now endogenous in our model.
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be the same since there are no
dded is paid to households).
ppens to the size of our multiplier effects?
overnment consumption increase.
nding. This further stimulates demand
refore larger.
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riculture
dustry
rvices
- agriculture
- industry
- services
or
ital
- rural
- urban
t
stment
world
row total
ficient matrix are exogenous accounts in our model.
se accounts as zero, since they do not generate
fects. All other columns should sum to one.
7/30/2019 Task 4 Solution
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commands for this exercise.
eps when entering the formula:
ant your answer to be (B67:N79).
y the coordinates for the entire identity matrix (B49:N61).
s for the entire coefficient matrix (B30:N42) (leave out the total row)
TRL) and SHIFT keys, and then press enter
the matrix to the left you should see that your formula
s around it (i.e., { and }). This is how Excel shows that
7/30/2019 Task 4 Solution
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