A Multiplier-Accelerator Input-Output Model
Óscar Dejuán & Ana González
Department of Economics and Finance. University of Castilla-La Mancha
Postal Address: Pz. Universidad, 1. 02071 Albacete (Spain)
E-mail: oscar.dejuan@uclm.es; anarosa.gonzalez@uclm.es
*Corresponding author
Abstract
Net output in a given period (as well as the employment and fixed capital required to
produce it) may be presented as a multiple of expected autonomous demand for the
same period. This is Keynes’ principle of effective demand encapsulated in the
multiplier model. Applying the same logic, the growth of output will be mostly
explained by the expected growth of autonomous demand.
The structural or disaggregated multiplier has been a basic tool of input-output
economics from its inception in the middle of the twenty century. The endogeneization
of final consumption of households has been approached by different methods.
Surprisingly enough, the attempts to endogeneize investment and to develop a
disaggregated multiplier-accelerator model are almost inexistent.
This will be the main contribution of our paper. After endogeneizing fixed capital
consumption, we shall try to derive net productive investment by firms from thee
elements: (a) the matrix of capital stocks; (b) the sectoral degree of capacity utilization;
and (c) the expected rate of growth in each sector that may be proxied by past rates of
growth.
The model can be used to figure out the evolution of income, employment and capital
stocks under different scenarios. We apply it to explain the actual dynamics of the
Spanish economy in the period 2005-08 and to predict its evolution from 2009 to 2012,
both in a pessimistic scenario (economic stagnation) and an optimistic one. Our
empirical analysis will be based on the symmetric input-output table of 2005 published
by INE (2009) and on the matrices of the stock of capital published by IVIE (2009).
Keywords: Input-Output
Multiplier, Accelerator.
Analysis;
Keynesian
Macroeconomics,
Investment,
Topic: 01: Methodological issues in input-output analysis; 09: Input-Output analysis
and structural change.
1. Introduction.
In 1787 Adam Smith set the agenda of Economics: “An inquiry into the nature
and causes of the Wealth of Nations”. Classical and Marxian economists devoted great
attention to the study of the accumulation of capital as the key explanatory variable of
economic growth (Marx, 1867). After a parenthesis of fifty years, Kalecki, Keynes and
their disciples (Keynes, 1936; Harrod, 1939; Kalecki, 1943; Domar, 1946) resumed the
interest for growth economics. They highlighted the double role of investment: it
increases the productive capacity of the economy in order to meet final demand and it is
nourishes final demand.
According to the Keynesian principle of effective demand, the output in a given
year is a multiple of the autonomous demand expected for the same period. The
multiplier results from endogeneizing final consumption which is a proportion (high and
stable) of disposable income. A part of investment (what we shall call “expansionary
investment”) may also be endogeneized given rise to the “supermultiplier” or
“multiplier-accelerator model” (Samuelson, 1939). Pasinetti (1974) considered that the
acceleration was the natural pattern of the multiplier in a macroeconomic model based
on the principle of effective demand. Amazingly enough, few economists (even among
the Keynesian tradition) have made full use of it.
The “matrix multiplier” has been a cornerstone of input-output analysis, from
different theoretical perspectives and for different purposes (Leontief, 1940; Goodwin,
1947; Miyazawa & Masegi, 1968; Kurz, 1985; Pyat & Round, 1985; Dejuán, Cadarso,
Córcoles, 1994). It has been profusely used to analyse the impact of a public work and
international trade patterns. It is also useful tool for growth analysis, although this
perspective has attracted less attention. By disaggregating the economy we can find out
the particular industries that play the role of locomotives in a given period.
Surprisingly enough, input-output analysts have not been interested in
disaggregating and endogeneizing the investment function in order to build a
“multiplier-accelerator model”.
The exception is Wasily Leontief who wrote the
“Dynamic Inverse” in 1970. The purpose of his paper had little relation with economic
growth in capitalist economies. He searched for the vector of fixed capital that will
make possible a given vector of consumption goods in year t and for the creation of the
required capital stock from now to year t. We know, however, that, in a capitalist
economy, consumption is clearly endogenous and investment (at least part of it) has an
autonomous nature.
Neither the modern Computable General Equilibrium (CGE)
models have paid attention to the investment function. They focus on the allocation of
income between consumption and savings, assuming that all savings will be invested.
(For a general exposition see: Ginsburgh & Keyzer, 2002. For a critical one see:
Dejuán, 2006)
In this paper we try to develop a sensible way to endogeneize a part of
productive investment and to integrate it into an input-output multiplier-accelerator
model. Our empirical support will be the last symmetric input-output table for the
Spanish economy corresponding to year 2005 (INE, 2009). The composition of the
stock of capital has been found in the data bank of IVIE (2009). To simplify the
exposition we have aggregated the table into 12 sectors (11 industries plus households).
The industries producing capital goods are three: 4: Vehicles; 5: Machinery and
equipment; 6: construction1.
The structure of the paper is as follows. Section 2 reminds the structure and
meaning of the aggregate multiplier-accelerator model. Section 3 builds an extended
input-output model after endogeneizing the bulk of final consumption of households
and fixed capital consumption of firms. Section 4 explains the dynamics of the system
output, employment and capital as a function of “proper autonomous demand”. The
first step consists in deriving productive investment of firms according to the principle
of acceleration.
With reference to the Spanish economy, section 6 simulates the
evolution of value added, employment and the capital stock under different scenarios.
Section 7 summarizes the conclusions.
2. The aggregate multiplier-accelerator model.
Kalecki (1943) and Keynes (1936) considered capitalism as a demandconstrained system, based on the principle of effective demand. According to this,
income, employment and the capital stock in a given year adjust to expected final
1
A fast look to tables 1 and 3 (below) may be useful to have an idea of the structure of the economy we
are considering.
demand. Net production in a given year can be presented as a multiple of autonomous
demand for the same period.
In the simplest model we can write the following
equation. (Since all variables refer to year t we’d better omit the temporal sub-index for
the time being).
Y  F1  C  F 2  C  I  F 3.
C  c·Y
I  k ·g ·Y
[2.1]
or
I  K ·g
Y stands for net output or income; F1 for net final demand (consumption, investment
and exports); F2 for autonomous demand (exports; and the part of consumption and
investment that does not depend on current income); F3 for proper autonomous demand
(exports, residential investment by households, modernization investment by firms,
public investment, public consumption and other forms of autonomous consumption). C
refers to induced final consumption, i.e. households’ expenditures that can be computed
as c times disposable income. The propensity to consume (c) has proved to be high and
stable. I refers to productive investment by firms of the expansionary type. The way it
has been formulated shows that firms adjust the stock of capital in order to attend
efficiently the expected increases in demand. K is the stock of capital at the beginning
of the production period. k is the optimal or desired “capital/output” ratio. g is the
expected rate of growth of the economy which, by construction, coincides with the
growth of proper autonomous demand.
We can present income as a multiple of proper autonomous demand by means of
the multiplier-accelerator relationship (the “supermultiplier”, so to speak):
Y
1
·F 3
1  c  k ·g
[2.2]
The problem with the supermultiplier is that it includes and fixes a variable (the
expected rate of growth) that is quite volatile. If the stock of capital is known we can
compute investment by an ad hoc procedure and introduce the result in autonomous
demand (F2=F3+I). This allows us to use the simple multiplier relationship: =1/(1-c).
Y
1
1
·F 2 
·I  F 3
1 c
1 c
[2.3]
All the variables refer to the current period of production (t). In the uncertain
horizon that defines capitalist economies, firms tend to proxy the expected growth of
demand for the current year and for the future, by the rate of growth registered in the
past year. Errors will show up as an excess or a lack of capacity. Latter on, these
excesses and shortages will be subtracted or added to the investment decided according
to the accelerator principle. The formula would look like this one:
I (t )  ao  K (t ) g (t 1) ·(1  a1 ·DU (t 1) )  
[2.4]
I(t) refers to investment expenditures by firms at the end of period t (31 December, so to
speak). K(t) is value of capital stock at the beginning of the period (1 January). DU(t-1):
are the deviations of normal capacity utilization dragged from the previous period. ao
and a1 are the parameters to be estimated.  is the residual error, with the usual
properties.
The preceding equation has been checked for the Spanish economy (1980-2005)
with good results. All the estimated parameters have the expected sign (positive) and
are statistically relevant; R2 amounts to 0,7. The goodness of fit is not as brilliant as the
consumption function which presents a R2=0,95 (being the propensity to consume
c=0,8). Yet it is much better than other investment functions which emphasize the role
of interest rates. As far as we know, all the students who have approached investment
empirically have concluded that the “flexible accelerator” is superior to any other
hypothesis. (Epstein & Denny, 1983; Andrés, Escribano, Molina y Taguas, 1990;
Raymond, Maroto y Melle, 1999; Kenny & Williams, 2001; Baddeley, 2003).
3. The extended input-output model.
Endogeneization of final
consumption and fixed capital consumption.
In this section we’ll widen the ordinary transactions table in order it includes the
endogenous part of final consumption of households and the purely endogenous part of
gross investment that coincides with fixed capital consumption.
The endogeneization of final consumption requires a new column and a new
row. They will be our “sector 12”. The 12th column of table 2 gathers endogenous
consumption by households that we shall interpret in a broad sense. It will include all
consumption expenditures except those of tourists that we shift to the export column. In
the 12th row we have to include the incomes that finance final consumption. The
coincidence of the total values of column and row 12th implies that the household sector
does not generate value added. It is supposed to produce a basket of consumption
goods to feed households. This basket will be financed by the bulk of wages and a part
of profits.
A proper endogeneization of consumption requires building a social accounting
matrix (SAM) which links primary incomes (i.e. factor incomes), with the disposable
income of institutions. A short-cut seems possible after verifying that the ratio “final
consumption of domestic households / value added in the economy” keeps relatively
constant (around 63%) in the last decade. The short-cut consists in considering that
63% of factor income is devoted to finance induced consumption.
The endogeneization of fixed capital consumption (FCC) by firms requires the
following steps. To begin with, we observe that, in the aggregate and for the last
decade, the proportion of FCC in gross investment (GI) amounts to 52%.
FCC(ag)=0,52·GI (ag means “aggregated”)
The FCC so computed is allocated among the three capital goods and among
sectors according to the weight of each capital good (Ki) in the total stock of capital (K).
We also consider the speed of depreciation of each capital good, (an inverse measure of
the number of years that each capital good is considered to endure, ni)2. From the
following formula we compute the parameter a’ which ensures that the whole value of
FCC is allocated into our 12 sectors.
K 1 K 1 K 1 
FCC(ag )   1 ·  2 ·  3 · ·a'
 K n1 K n2 K n3 
FCC(ag )
a' 
(.)
[3.1]
Each cell of the FCC matrix is computed multiplying a’ by the capital share
corresponding to each sector and good.
2
According to IVIE (2009) the amortization period reaches 14 years for vehicles (industry 3); 11,25 years
for machinery and equipment (industry 4); 44,28 years for industrial constructions; and 60 years for
dwellings owned by households.
 K 31 1
· ·a '

 K 3 n3
K 1
FCC   41 · ·a '
K n
 4 4
 K 51 · 1 ·a '
 K 5 n5

K 32 1
· a'...
0

K 3 n3

K 42 1

· ·a '...
0

K 4 n4

K 5,12 1 
K 52 1
· ·a '...
· ·a '
K 5 n5
K 5 n5 
[3.2]
FCC is a 12·12 matrix, although only the rows corresponding to industries producing
capital goods (3, 4 and 5) are filled. All the cells of the households sector (12th column)
are nil except the one corresponding to the construction industry (5th row). There we
include the depreciation of houses owned by families.
Adding up FCC to the previous transaction table (which already includes
endogenous final consumption) we obtain the extended matrix of “circulating capital”
(CC). It includes intermediate consumption, fixed capital consumption and induced
final consumption. A multiplier model is mostly interested in the circulating capital
produced in the country (CCd).
It is obtained by subtracting imports from total
transactions. (The information is provided by the original TIO which distinguish among
total, domestic and imported quantities).
In the next three tables, all them referred to the Spanish Economy in 2005, we
present empirical information about the matrix of capital stocks (table 1), the matrix
fixed capital consumption (table 2) and the modified input output matrix that will serve
as the starting point of our analysis (table 3)
Table 1. Matrix of capital stocks (million Euros). (KI)
1.Agriculture
4.Vehicles
3.Intermediate
Goods
2.Energy
4.Vehicles
5.Machinery
6.Construction
7.Consumption
Goods
8.Transport
9.Restauration
10.Market
services
11.Non
Mark. Serv.
12. Households
5.618
397
2.189
1.470
417
5.011
1.853
77.420
612
35.769
6.904
5.Machinery
21.836
40.010
55.554
20.352
14.813
23.126
50.855
41.131
14.846
68.950
56.655
0
0
6.Construction
71.888
93.217
130.819
16.649
20.289
76.691
98.825
179.761
44.416
423.780
497.443
2.117.526
Total
99.342
133.624
188.562
38.471
35.519
104.828
151.533
298.312
59.873
528.499
561.002
2.117.526
Table 2. Fixed capital consumption (million Euros). (FCC)
1.Agriculture
4.Vehicles
3.Intermediate
Goods
2.Energy
4.Vehicles
5.Machinery
6.Construction
7.Consumption
Goods
8.Transport
9.Restauration
10.Market
services
11.Non
Mark. Serv.
12.
Households
346
24
135
90
38
451
114
4.611
38
2.202
425
5.Machinery
1.673
3.065
4.256
1.559
1.557
2.431
3.896
2.069
1.137
5.282
4.340
0
0
6.Construction
1.399
1.814
2.546
324
973
3.678
1.923
734
864
8.247
9.680
30.414
Total
3.417
4.903
6.936
1.973
2.567
6.560
5.933
7.414
2.039
15.730
14.445
30.414
Table 3. Modified IOT 2005 (million Euros).
1.Agriculture
3.Intermediate
Goods
2.Energy
4.Vehicles
5.Machinery
6.Construction
7.Consumption
Goods
8.Transport
9.Restauration
10.Market
services
11.Non
Mark. Serv.
12.
Households
1.Agriculture
2.316
1
996
2
0
23
24.296
9
1.436
1.328
174
9.400
2.Energy
1.155
42.016
9.545
730
1.885
1.030
3.199
6.532
1.149
11.307
3.099
15.134
3.Intermediate Goods
1.680
230
43.059
9.608
25.565
32.089
13.084
339
1.820
6.791
3.396
7.801
14.372
4.Vehicles
349
27
226
22.040
313
451
165
5.177
80
8.397
567
5.Machinery
3.101
4.653
15.077
5.929
26.878
23.661
9.682
3.368
2.230
14.415
8.353
9.088
6.Construction
7.Consumption
Goods
1.604
2.302
3.203
399
1.256
101.626
2.809
1.016
2.064
27.815
11.505
34.774
6.449
308
3.099
826
758
2.614
39.031
570
19.123
19.549
3.707
66.118
249
769
6.915
854
1.998
1.909
6.570
1.118
118
13.381
1.457
6.188
15
74
367
59
134
247
168
123
61
4.850
1.242
74.931
3.716
6.801
22.164
6.056
13.873
23.925
28.334
18.408
14.723
151.086
21.215
243.069
0
0
0
0
0
0
0
0
0
0
0
2.146
16.947
11.596
24.822
6.454
20.589
59.867
27.837
15.004
39.404
246.620
68.358
0
8.Transport
9.Restauration
10.Market services
11.Non Mark. Serv.
12. Households
Rest of added value
Production
5.168
972
5.640
1.296
7.864
23.771
8.171
187
17.924
109.219
20.188
0
40.822
70.806
136.077
54.353
101.126
273.666
160.770
54.357
101.422
627.912
147.726
537.496
13.Intermediate
demand (1 -12)
14.Net fixed
capital
formation
15.Public
expenditures
1.Agriculture
39.981
0
0
8.005
813
8.818
48.799
2.Energy
96.781
0
0
8.027
276
8.303
105.084
145.461
0
6.856
38.489
-221
45.125
190.586
52.164
6.026
222
33.230
65
39.543
91.707
5.Machinery
126.434
4.867
595
31.676
86
37.223
163.658
6.Construction
7.Consumption
Goods
190.372
73.022
10.279
9
0
83.310
273.682
162.151
0
7
33.478
4.562
38.047
200.198
8.Transport
41.527
0
1.365
17.275
217
18.857
60.384
9.Restauration
82.271
0
0
20.189
0
20.189
102.460
553.369
0
17.303
46.085
42.401
105.789
659.158
147.726
3.Intermediate Goods
4.Vehicles
10.Market services
11.Non Mark. Serv.
12. Households
Total
16.Total
Exports
18.Final
demand
(14-17)
17.Others
19.Total
output
=13+18
2.146
0
137.545
28
8.007
145.581
537.496
0
0
0
0
0
537.496
2.030.153
83.915
174.171
236.492
56.206
550.784
2.580.937
We proceed now to compute the traditional matrices of technical coefficients
that will be the basic tool for our analysis.
Ad*  CC d ·q 1
v  VAN ·q 1
l  L·q 1
k  KI ·q
[3.3]
1
z d  FCC·q 1
q is the column vector of total output. VAN is the row vector of net value added (for our
purposes it is convenient to use the original figures of the symmetric IOT, i.e. the sum
of wages and profits). L is a row vector that indicates the number of workers employed
in each industry. KI is a (3·12) matrix gathering the stock of capital goods installed in
each sector. A* is the extended technical matrix. l is the row vector of labour
coefficients; k: is the matrix of capital coefficients. We suppose that the original TIO
reflects an economic equilibrium so the capital coefficients derived from it, stand for the
optimal or desired “capital/output” ratios. zd is a (3·12) matrix gathering fixed capital
consumption per unit of output. (Tables 4, 5 & 6 show the values of these vectors and
matrices in the Spanish economy)
The Leontief’s inverse corresponding to matrix A*d is the multiplier of total
output (Mq). From here we obtain the multiplier of income or net value added (Mv),
the multiplier of employment (Ml) and the multiplier of fixed capital (Mk). (See tables
7, 8, 9 and 10)

Mq  I  Ad*

1


Ml  l ·I  A 
Mk  k ·I  A 
Mv  v· I  Ad*
1
[3.4]
* 1
d
* 1
d
Table 4. Unit value added ( v).
1.Agriculture
v
0,625
2.Energy
0,247
3.Intermediate
Goods
0,275
4.Vehicles
0,179
5.Machinery
6.Construction
7.Consumption
Goods
0,307
0,330
0,261
8.Transport
9.Restauration
0,416
0,585
10.Market
services
0,592
11.Non
Mark. Serv.
0,697
12.
Households
0,000
Table 5. Vector of labour coefficients (l).
1.Agriculture
l
0,024
2.Energy
0,001
3.Intermediate
Goods
0,006
4.Vehicles
0,004
5.Machinery
6.Construction
7.Consumption
Goods
0,008
0,009
0,008
8.Transport
0,012
9.Restauration
0,013
10.Market
services
0,014
11.Non
Mark. Serv.
0,018
12.
Households
0,000
Table 6. Matrix of capital coefficients (k).
1.Agriculture
2.Energy
3.Intermediate
Goods
4.Vehicles
5.Machinery
6.Construction
7.Consumption
Goods
8.Transport
9.Restauration
10.Market
services
11.Non
Mark. Serv.
12.
Households
4.Vehicles
0,138
0,006
0,016
0,027
0,004
0,018
0,012
1,424
0,006
0,057
0,047
0,000
5.Machinery
0,535
0,565
0,408
0,374
0,146
0,085
0,316
0,757
0,146
0,110
0,384
0,000
6.Construction
1,761
1,317
0,961
0,306
0,201
0,280
0,615
3,307
0,438
0,675
3,367
3,940
Total
2,434
1,887
1,386
0,708
0,351
0,383
0,943
5,488
0,590
0,842
3,798
3,940
11.Non
Mark. Serv.
12.
Households
Table 7. Multiplier of total output (Mq).
1.Agriculture
2.Energy
3.Intermediate
Goods
4.Vehicles
5.Machinery
6.Construction
7.Consumption
Goods
8.Transport
9.Restauration
10.Market
services
1.Agriculture
1,124
0,025
0,044
0,026
0,034
0,045
0,205
0,043
0,090
0,051
0,051
0,068
2.Energy
3.Intermediate
Goods
0,125
1,300
0,152
0,077
0,103
0,103
0,122
0,190
0,104
0,107
0,113
0,119
0,151
0,064
1,278
0,210
0,277
0,306
0,182
0,101
0,123
0,104
0,117
0,112
4.Vehicles
0,022
0,010
0,018
1,187
0,018
0,019
0,020
0,073
0,018
0,030
0,020
0,021
5.Machinery
0,136
0,084
0,158
0,094
1,216
0,203
0,128
0,102
0,092
0,080
0,100
0,082
6.Construction
7.Consumption
Goods
0,298
0,183
0,217
0,136
0,188
1,803
0,246
0,242
0,257
0,287
0,347
0,314
0,401
0,111
0,174
0,120
0,151
0,202
1,423
0,192
0,426
0,228
0,231
0,291
8.Transport
0,062
0,040
0,093
0,048
0,064
0,066
0,095
1,067
0,057
0,067
0,057
0,062
9.Restauration
0,172
0,088
0,121
0,083
0,116
0,153
0,141
0,147
1,163
0,165
0,178
0,264
10.Market services
11.Non Mark.
Serv.
1,033
0,591
0,861
0,599
0,798
0,984
1,018
1,132
1,042
2,084
1,045
1,273
0,005
0,002
0,003
0,002
0,003
0,004
0,004
0,004
0,004
0,004
1,005
0,007
12. Households
1,171
0,590
0,800
0,554
0,776
1,031
0,950
0,981
1,106
1,083
1,160
1,829
Total
4,699
3,087
3,919
3,137
3,745
4,920
4,533
4,274
4,481
4,290
4,424
4,442
Table 8. Multiplier of income or net value added (Mv).
1.Agriculture
2.Energy
3.Intermediate
Goods
4.Vehicles
5.Machinery
6.Construction
7.Consumption
Goods
8.Transport
9.Restauration
10.Market
services
11.Non
Mark. Serv.
12.
Households
1.Agriculture
0,703
0,015
0,027
0,016
0,022
0,028
0,128
0,027
0,056
0,032
0,032
0,042
2.Energy
3.Intermediate
Goods
0,031
0,321
0,037
0,019
0,025
0,026
0,030
0,047
0,026
0,026
0,028
0,029
0,041
0,018
0,351
0,058
0,076
0,084
0,050
0,028
0,034
0,029
0,032
0,031
4.Vehicles
0,004
0,002
0,003
0,212
0,003
0,003
0,004
0,013
0,003
0,005
0,004
0,004
5.Machinery
0,042
0,026
0,048
0,029
0,373
0,062
0,039
0,031
0,028
0,025
0,031
0,025
6.Construction
7.Consumption
Goods
0,098
0,060
0,071
0,045
0,062
0,594
0,081
0,080
0,085
0,095
0,114
0,103
0,105
0,029
0,045
0,031
0,039
0,053
0,371
0,050
0,111
0,059
0,060
0,076
8.Transport
0,026
0,017
0,039
0,020
0,027
0,028
0,040
0,444
0,024
0,028
0,024
0,026
9.Restauration
0,101
0,052
0,071
0,049
0,068
0,090
0,083
0,086
0,681
0,097
0,104
0,155
10.Market services
11.Non Mark.
Serv.
0,611
0,350
0,510
0,354
0,472
0,582
0,602
0,670
0,617
1,233
0,618
0,753
0,003
0,002
0,002
0,002
0,002
0,003
0,003
0,003
0,003
0,003
0,700
0,005
12. Households
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
Total
1,765
0,890
1,206
0,835
1,169
1,553
1,431
1,478
1,667
1,632
1,748
1,250
Table 9.Multiplier of employment (Ml)
1.Agriculture
3.Intermediate
Goods
2.Energy
4.Vehicles
5.Machinery
6.Construction
7.Consumption
Goods
8.Transport
9.Restauration
10.Market
services
11.Non
Mark. Serv.
12.
Households
1.Agriculture
0,024
0,001
0,001
0,001
0,001
0,001
0,004
0,001
0,002
0,001
0,001
0,001
2.Energy
0,000
0,001
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
3.Intermediate Goods
0,001
0,000
0,007
0,001
0,001
0,002
0,001
0,001
0,001
0,001
0,001
0,001
4.Vehicles
0,000
0,000
0,000
0,005
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
5.Machinery
0,001
0,001
0,001
0,001
0,009
0,002
0,001
0,001
0,001
0,001
0,001
0,001
6.Construction
7.Consumption
Goods
0,003
0,002
0,002
0,001
0,002
0,016
0,002
0,002
0,002
0,003
0,003
0,003
0,003
0,001
0,001
0,001
0,001
0,001
0,010
0,001
0,003
0,002
0,002
0,002
8.Transport
0,001
0,000
0,001
0,001
0,001
0,001
0,001
0,012
0,001
0,001
0,001
0,001
9.Restauration
0,002
0,001
0,001
0,001
0,001
0,002
0,002
0,002
0,014
0,002
0,002
0,003
10.Market services
0,012
0,007
0,010
0,007
0,009
0,011
0,012
0,013
0,012
0,024
0,012
0,015
11.Non Mark. Serv.
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,018
0,000
12. Households
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
Total
0,047
0,013
0,025
0,018
0,026
0,036
0,034
0,033
0,035
0,034
0,040
0,027
Table 10.Multiplier of fixed capital (Mk).
1.Agriculture
3.Intermediate
Goods
2.Energy
4.Vehicles
5.Machinery
6.Construction
7.Consumption
Goods
8.Transport
9.Restauration
10.Market
services
11.Non
Mark. Serv.
12.
Households
1.Agriculture
0
0
0
0
0
0
0
0
0
0
0
0
2.Energy
3.Intermediate
Goods
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4.Vehicles
0,317
0,108
0,218
0,147
0,158
0,200
0,248
1,603
0,173
0,234
0,209
0,184
5.Machinery
1,101
0,949
0,918
0,766
0,592
0,643
0,956
1,249
0,663
0,555
0,833
0,516
6.Construction
7.Consumption
Goods
8,261
4,858
5,817
3,635
4,727
6,212
6,489
8,885
6,544
6,559
9,543
9,084
0
0
0
0
0
0
0
0
0
0
0
0
8.Transport
0
0
0
0
0
0
0
0
0
0
0
0
9.Restauration
0
0
0
0
0
0
0
0
0
0
0
0
10.Market services
11.Non Mark.
Serv.
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
12. Households
Total
0
0
0
0
0
0
0
0
0
0
0
0
9,679
5,915
6,953
4,548
5,478
7,056
7,693
11,737
7,381
7,348
10,585
9,783
4. Expansionary investment and the dynamics of the system.
In our disaggregated Keynesian model income, employment and capital in a
given moment of time can be presented as a multiple (supermultiplier) of the vector of
proper autonomous demand (F3) expected for this year.
For the same logic, the
evolution of these variables through time will be linked to the expected growth of the
goods included in vector F3.
Proper autonomous demand includes net residential investment by households,
net investment in infrastructures by government and other types of net investment
unrelated to income. It does not include expansionary investment which is supposed to
depend on the expected growth of income.
Its exact value may be computed
multiplying the expected rate of growth of the economy for the capital installed. (K).
I (12·1),(t )  K (12·12),(t ) ·g (e1·12),(t )
[4.1]
I(t) is the investment column vector at the end of period t. K(t) informs about the stocks
of each capital good in each sector at the beginning of period t. ge stands for the
expected rate of growth of each commodity. In the usual uncertainty that characterizes
private decisions, the expected growth for the current year (t) is proxied by the actual
rate of growth of the economy in the previous period (g(t-1)). Errors will show up in
excesses of capacity (positive or negative) (EK(t-1)). Later on, they will be subtracted or
added to the investment derived form the acceleration principle (K(12·12)(t)·g(12·1)(t-1)) in
order to approach the desired “capital/output” ratio in each sector.
I (12·1),( t )  K (12·12),( t ) ·g (12·1),( t 1)  EK (12·1),( t 1)
K (12·12),( t )  K (12·12),( t 1)  I (12·12),( t 1)
EK (12·1),( t 1)  KR(12.12),(t 1) ·i(12·1)  K (12·12),( t 1) ·i(12·1)
[4.2]
KR(12·12),( t 1)  k (12·12), ·q ( t 1) · g (t 1)
EK(t-1) gathers the excesses of capacity. It results from subtracting installed
capital in period (t-1)(K), from required capital in the same period (KR). (Both stocks
appear in matrix form; i is a column vector of ones which adds up the value of the rows
of capital matrices). Required capital (KR(t-1)) results from multiplying the desired
capital/output ratios (k) times net output in (t-1), times effective rates of growth of
sectoral output in (t-1).
Installed capital in any year t results from adding net
investment at the end of period t to the stock of capital installed at the beginning of the
same period (see the second equation of [4.2], where net investment is presented in
matrix form)
As we have said the level of net output in year t will be a multiple of the
expected autonomous demand (F2), a part of which consists in capital goods to expand
productive capacity (I) and the rest corresponds to “proper autonomous demand” (F3).

Y(1,12),(t )  v· I  Ad*
 ·I
1
(12·1),( t )
 F 3(12·1),(t ) 
[4.3]
Suppose F3 is growing a <g>. We present it as a diagonal matrix and allow differences
in sectoral rates of growth. These rates are known. Output in the next year (t+1) will
be:

Y(1.12),(t 1)  v· I  Ad*
 ·I
1
(12·1),( t 1)
 g
(12·12),( t )

·F 3(12·1),(t ) .
[4.4]
The dynamics of employment and capital is obtained by a similar formula, using
the corresponding multiplier. For any year t+1 we can derive income, employment and
capital from the vector of proper autonomous demand in the base year (t) and the rate of
growth of each element of F3 in year t.

L(1,12),(t 1)  l · I  Ad*

 ·I
1
K (12,12),(t 1)  k · I  Ad*
(12·1), ( t 1)
 ·I
 g
1
(12·1), ( t 1)
(12·12),( t )
 g
·F 3(12·1),(t )
(12·12), ( t )

F 3(12·1),(t )
[4.5]

[4.6]
5. Simulations of the dynamics of income, employment and capital in
the Spanish economy.
In this section we shall apply our multiplier-accelerator model to simulate the
evolution of the Spanish economy during the period 2005-2012. We know the actual
evolution of final demand from 2005 to 2008. We do not use, however, all available
information because our interest lies in predicting future outcomes with limited
information. An example. To compute sectoral net productive investment we do not
use the rates of growth of each sector in the current period but in the previous rate.
Errors will be corrected in the following year. Estimated investment in 2008 will be
higher than the actual one because our model considers the rates of growth in 2007,
when the economy was booming. Of course, entrepreneurs did cut investments in the
second part of 2008, as soon as they appreciated that the economy had entered into a
deep recession. Yet, a model used for prediction cannot foresee these changes; it is
bound to look backwards and needs some time to correct errors.
For year 2009 till 2012 we have considered three possible scenarios. In the
pessimist scenario the proper autonomous demand (F3) falls 6% in 2009 and remains
stagnant from 2010 to 2012. In the first optimistic scenario F3 is almost stagnant in
2009 and resumes growth in 2010 (+2,5% in 2010; +3% in 2011 and 2012). The second
optimistic scenario is similar to the previous one but there is an element of autonomous
demand (construction) growing at 6% (as it was the case of the years previous to the
recession).
Figure 1 shows the diverging patterns of value added after 2009.
In the
pessimistic scenario the level of aggregate VA coincides with the initial one (2005). In
the optimistic scenario VA in 2012 is slightly over the 2007-08 peak.
Figure 1. Estimation of value added.
2005 = 100
160
140
120
OPTIMISTIC SCENARIO
100
80
OPTIMISTIC SCENARIO*
PESSIMISTIC SCENARIO
60
40
ACTUAL DATA
20
0
2005 2006 2007 2008 2009 2010 2011 2012
t
In figure 2 we observe the evolution of employment in the three scenarios. In
figures 3.a, 3.b and 3c we observe the evolution of the stocks of vehicles, machinery
and buildings. Notice that in the pessimistic scenario the stock of capital does not fall.
This is due to the asymmetric nature of the accelerator that does not allow for negative
investments.
destroyed.
In recessions a part of installed capital remains idle but cannot be
2005 = 100
Figure 2. Estimation of employment.
160
140
120
100
80
60
40
20
0
OPTIMISTIC SCENARIO
OPTIMISTIC SCENARIO*
PESSIMISTIC SCENARIO
ACTUAL DATA
2005
2006
2007
2008
2009
2010
2011
2012
t
Figure 3.a. Estimation of the stock of vehicles.
125
2005=100
120
115
OPTIMISTIC SCENARIO
110
OPTIMISTIC SCENARIO*
105
PESSIMISTIC SCENARIO
100
ACTUAL DATA
95
90
2005
2006
2007
2008
2009
2010
2011
2012
t
Figure 3.b. Estimation of the stock of machinery.
115
2005 = 100
110
OPTIMISTIC SCENARIO
105
OPTIMISTIC SCENARIO*
100
PESSIMISTIC SCENARIO
ACTUAL DATA
95
90
2005
2006
2007
2008
2009
t
2010
2011
2012
Figure 3. c. Estimation of the stock of construction.
160
140
2005 = 100
120
OPTIMISTIC SCENARIO
100
OPTIMISTIC SCENARIO*
80
PESSIMISTIC SCENARIO
60
ACTUAL DATA
40
20
0
2005
2006
2007
2008
2009
2010
2011
2012
t
How accurate are the predictions of our model? We can answer by comparing the
estimation of value added, employment or capital stocks with the actual data that we
know for years 2006, 2007 and 2008. Figure 5, as an example, shows graphically the
differences between the patterns of VA estimated and real.
Table 11 computes with more detail the errors of prediction of VA in years
2006-08. They reflect the difference between the estimated rates of variation of value
added (E^e) and the actual rates (E^). The last cell computes the MAE (means of
absolute errors) for the whole period (3 years), according to the following formula:
MAE i 
1 t ˆe
 Ei,t  Eˆ i,t
3 1
[5.1]
The usual threshold for acceptance of errors and MAEs is 5%. In our case the
MAE amount to 2,95%, which is quite good, indeed. As a matter of fact errors of
prediction for years 2006 and 2008 are much lower than the errors corresponding to
year 2008. It is not surprising because 2008 is the turning point the economy (from
boom to bust). Our model will correct the error of 2008 in the following year cutting
investment even more that the contraction of the economy requires.
Figure 4. VA estimated and real.
Millions Euros
1000000
950000
900000
REAL
850000
ESTIMATION
800000
750000
700000
2005
2006
2007
t
Table 11. Errors of prediction.
Estimated
rate of
change of
value added
(%)
2006
9,851
2007
7,106
2008
0,363
Actual rate
of change of Errors of
value added prediction
(%)
(%)
MAE (%)
4,364
5,488
4,872
2,234
1,481
-1,118
2,947
6. Conclusions.
The purpose of the paper was to develop a multiplier-accelerator model in an
input-output framework. To achieve this result we have endogeneized the bulk of final
consumption of households, following traditional procedures. We have also
endogeneized and included in the transaction matrix, the part of gross investment
corresponding to fixed capital consumption. This stands for the first methodological
novelty of the paper.
Net productive investment has not been introduced into the transaction matrix
because it depends on the rate of growth of autonomous demand, a parameter that is
quite volatile.
Although we have kept net “expansionary” investment in the
“multiplicand”, we have computed it “ad hoc”.
Proper autonomous demand (the
locomotive of the system) includes exports, public expenditures, residential investment
of households and modernization investment of firms. “Expansionary” investment
depends on the expected rate of growth of output and can be computed multiplying the
capital stock installed in each sector by the past rate of growth of the sector. Errors will
be reflected in an excess or a lack of capacity and will be subtracted or added to the
investment decisions based on the acceleration principle.
This is the second, and
probably most important, contribution of the paper.
The model has been applied to explain the evolution of value added,
employment and capital in the Spanish economy from 2005 to 2008 and to forecast the
dynamics of the same variables from 2009 to 2012. In the forecasting exercise we have
considered three possible scenarios. In the pessimist one, the economy experiences a
big fall in 2009 and remains stagnant in the following years. In the optimistic ones the
fall is smaller and the recovery starts in 2010. The results are interesting because apart
from the estimation of aggregate employment we can visualize the sectors that will gain
more or that will loss more.
The fact that we have the actual data for years 2005-2008 allows evaluating the
accuracy of the model. Since predictions errors are well below the 5% threshold we are
allowed to conclude that the model predicts reasonable well.
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