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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: [email protected]; [email protected] *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. 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