Input-Output Model and its Application
3.8 Varieties of additional concepts on input-output modeling
3.8.8 Alternative Methods to estimate multipliers
There are alternative concepts to understand how the multipliers work and how they can be estimated. Here two methods are explained.
3.8.8.1 Using Keynesian multiplier formula
Some of you might have seen the formula using the marginal propensity to consume (MPC) in a standard tourism textbook (Goeldner and Ritchie, 2006).
Multiplier can be estimated as follows:
Multiplier
1
1 MPC (3.17)
Where M is marginal, P is propensity, and C is consume.
MPC
Consumption
Disposable income (3.18)
A tourist group spends $10 000 in the study region. The MPC has been estimated to be 0.5, which assumes that host region ’ s industrial sectors and local people tend to spend 0.5, or half, of whatever they receive as additional revenues or income.
Then, in addition to initial expenditure by the tourist of $10 000 will yield the following impacts by way of sequencing rounds of new expenditures as follows ( Table 3-14 ).
As you can see in Table 3-14 , with the MPC of 0.5, the initial expenditure of $10 000 results in the accumulated impact of $20 000, which yields the multiplier of 2. MPC and the resulting multiplier appear to have the multiplicative inverse relationship. Let us see if that holds with two other cases: case 2 with an MPC of 0.7 and case 3 with an MPC of 0.3 ( Table 3-16 ).
By considering cases 2 and 3, MPCs and the resulting multipliers do not hold a multi- plicative inverse relationship. What you can see is the substantial difference in multipliers in response to a small change in parameters. If you look at the sum of all the responding rounds in three cases, excluding the identical initial impact of $10 000, case 1 was $10 000, case 2 was
$23 324.21, and case 3 was $4285.71. While this method remains versatile and useful, we are reminded of its sensitivity to one single parameter. Marginal propensity to save (MPS) is sim- ply the remaining part of the income after you spent some portion. So, MPS 1 MPC.
MPS
Savings
Disposable income (3.19)
Saving is considered as leakage to the regional economy, as you removed the money from circulation. Another leakage is import. When you import goods and services, money flows
I N P U T-O U T P U T M O D E L A N D I T S A P P L I C AT I O N 97
Table 3-19 Keynesian estimation of multipliers with MPC 0.5 as a parameter (case 1) . Case 1 Parameter Table MPC 0.5
Additional Impact per round Accumulated Impacts Relative Size of Response
0 round $10 000.00 $10 000.00 1
1 round $5 000.00 $15 000.00 1.5
2 rounds $2 500.00 $17 500.00 1.75
3 rounds $1 250.00 $18 750.00 1.875
4 rounds $625.00 $19.375.00 1.9375
5 rounds $312.50 $19 687.50 1.96875
6 rounds $156.25 $19 843.75 1.984375
7 rounds $78.13 $19 921.88 1.992188
8 rounds $39.06 $19 960.94 1.996094
9 rounds $19.53 $19 980.47 1.998047
10 rounds $9.77 $19 990.23 1.999023
11 rounds $4.88 $19 995.12 1.999512
12 rounds $2.44 $19 997.56 1.999756
13 rounds $1.22 $19 998.78 1.999878
14 rounds $0.61 $19 999.39 1.999939
15 rounds $0.31 $19 999.69 1.999969
16 rounds $0.15 $19 999.85 1.999985
17 rounds $0.08 $19 999.92 1.999992
18 rounds $0.04 $19 999.96 1.999996
19 rounds $0.02 $19 999.98 1.999998
20 rounds $0.01 $19 999.99 1.999999
21 rounds $0.00 $20 000.00 2
22 rounds $0.00 $20 000.00 2
in the opposite direction from the local economy to outside of the local economy. So another version of the estimation of multiplier would be:
Multiplier (with leakage of imports considered)
1 1
( MPC MPII) (3.20)
where
MPImarginal propensity to import
Imports
Disposable incomme (3.21) For example, with MPC 0.5, the previous case showed the multiplier of 2. Now let us assume that MPI 0.1.
Then the multiplier with import leakage considered would be 1
1
1
1 0 5 0 1 1 667
( ) ( . . ) .
MPC MPI
Table 3-20 Keynesian estimation of multipliers with MPC 0.7 as a parameter (case 2; a) and MPC 0.3 as a parameter (case 3; b)
(a) Case 2 Parameter Table MPC 0.7
Additional Impact per round Accumulated Impacts Relative Size of Response
0 round $10 000.00 $10 000.00 1
1 round $7 000.00 $17 000.00 1.7
2 rounds $4 900.00 $21 900.00 2.19
3 rounds $3 430.00 $25 330.00 2.533
4 rounds $2 401.00 $27 731.00 2.7731
5 rounds $1 680.70 $29 411.70 2.94117
6 rounds $1 176.49 $30 588.19 3.058819
7 rounds $823.54 $31 411.73 3.141173
8 rounds $576.48 $31 988.21 3.198821
9 rounds $403.54 $32 391.75 3.239175
10 rounds $282.48 $32 674.22 3.267422
11 rounds $197.73 $32 871.96 3.287196
12 rounds $138.41 $33 010.37 3.301037
13 rounds $96.89 $33 107.26 3.310726
14 rounds $67.82 $33 175.08 3.317508
15 rounds $47.48 $33 222.56 3.322256
16 rounds $33.23 $33 255.79 3.325579
17 rounds $23.26 $33 279.05 3.327905
18 rounds $16.28 $33 295.34 3.329534
19 rounds $11.40 $33 306.74 3.330674
20 rounds $7.98 $33 314.72 3.331472
21 rounds $5.59 $33 320.30 3.33203
22 rounds $3.91 $33 324.21 3.332421
(b) Case 3 Parameter Table MPC 0.3
Additional Impact per round Accumulated Impacts Relative Size of Response
0 round $10 000.00 $10 000.00 1
1 round $3 000.00 $13 000.00 1.3
2 rounds $900.00 $13 900.00 1.39
3 rounds $270.00 $14 170.00 1.417
4 rounds $81.00 $14 251.00 1.4251
5 rounds $24.30 $14 275.30 1.42753
6 rounds $7.29 $14 282.59 1.428259
7 rounds $2.19 $14 284.78 1.428478
8 rounds $0.66 $14 285.43 1.428543
9 rounds $0.20 $14 285.63 1.428563
10 rounds $0.06 $14 285.69 1.428569
11 rounds $0.02 $14 285.71 1.428571
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(b) Case 3 Parameter Table MPC 0.3
Additional Impact per round Accumulated Impacts Relative Size of Response
12 rounds $0.01 $14 285.71 1.428571
13 rounds $0.00 $14 285.71 1.428571
14 rounds $0.00 $14 285.71 1.428571
15 rounds $0.00 $14 285.71 1.428571
16 rounds $0.00 $14 285.71 1.428571
17 rounds $0.00 $14 285.71 1.428571
18 rounds $0.00 $14 285.71 1.428571
19 rounds $0.00 $14 285.71 1.428571
20 rounds $0.00 $14 285.71 1.428571
21 rounds $0.00 $14 285.71 1.428571
22 rounds $0.00 $14 285.71 1.428571
Again, the concept of this method is very useful, and this method is actually easier to use.
It gets a little more challenging to use this method only when you have an access the rigorous I-O data of the study region, from which you can actually calculate various multipliers and the impact analyses for each industrial sectors.
3.8.8.2 Power series approximation of multipliers(I A ) 1
There is also an approximation of effect of multipliers without using the matrix computa- tions that we made. It is just a more mathematically intensive explanation than the previous Keynesian multiplier estimation. The discussions by Miller and Blair will be followed here.
Recall in matrix notations in which 0 A 1 (to be precise, all the elements in the A-matrix should be larger than 0: aij 0 for all i and j ).
If you remember how you created the A-matrix, you can agree that each of the column sums will be 1. Why can we say so? Each industrial sector purchases some amount of value added (such as labor, capital, and imports) that we disregarded in the process of creating A-matrix.
So we can say that:
aij j
i n <
∑
11
for all
(sum of all the coefficients from row 1 to row n along each column in the A-matrix is always smaller than 1, for all the columns)
Now let us consider some matrices equations.
(IA I)( A A2A3A4...An)
where, for square matrices, A2 AA , A3 AAA AA2 and they continue as such.
Now proceed with the above equation.
( )( ... )
...
I A I A A A A A
I A A A A A A A A A
n n
2 3 4
2 3 4 2 3 4 AA5... An1
It appears that you can simplify the equation by erasing and , or you match one with another with a different sign.
I A A A A A A A A A A A
I A A A A A
n n
2 3 4 2 3 4 5 1
2 2 3
... ...
A
A3A4A4A5A5...AnAn... An1 So, after simplifying the equation we have:
(IAn1)
Now let us assume that we have a very large n , such as infinity ( n→ ). The elements in the matrixAn 1 will all become zero. ( An1→ 0)
Now that An 1→ 0, then ( I An 1 ) will be ( I 0) I
Now, ( I A )( I A A2 A3 A4 … An ) 1, then by definition (I A) can become I only when it is multiplied by its multiplicative inverse, ( I A ) 1
(IA I)( A)1 I So we can say that ( I A ) 2 1 ( I A A2 A3 A4 … An ) Since ( I A )1 Y X , ( I A A2 A3 A4 … An ) Y X
Then remove the parenthesis, we will have Y AY A2Y A3Y A4Y … An Y X Y AY A ( AY ) A ( A2Y ) A ( A3Y ) … A ( An 2 1Y ) X
What we see here is that each term after the initial round is the preceding term multiplied by A. This is similar to normal algebra in which
1
1 1 2 3
( ) ... ,
a a a a an
for | a | 1.
Individual terms in the power series approximation show the round-by-round effects. This way, without using the Leontief inverse matrix, you can estimate the magnitude of effects.