A case study: the 1983 U.S. human capital

population of economic units. This combination offers a robust statistical support to the estimation of HC. The main objection to the prospective approach does not apply to our approach because it works with average earnings by age. These averages allow an accurate representation of the capacity or power of the average economic unit by age to produce a flow of earnings because, under general conditions of stochastic regularity, the law of large numbers applies. The applica- tion of the prospective method to the smoothed values of the average earnings by age allows the estimation of the weighted average of HC, i.e. the average HC of the population. Besides, this average becomes our scaling or benchmarking factor to pass from the transformed latent variable estimation to the monetary values and the size personal distribution of HC.

6. A case study: the 1983 U.S. human capital

The proposed new method of HC estimation presented in Section 4 allows, i the HC estimation of each economic unit as a latent variable; ii the average HC by age; iii the average HC of the population of economic units; and iv using the estimations obtained in i – iii, to pass from the HC estimation as a latent variable with zero mean and unit variance to the HC estimation in monetary values; and v to obtain from iv the size distribution of HC with mean given by iii. With the scope of testing the power and the validity of this approach, the method presented in Section 4, as outlined in Table 1 and in i – v above, is applied to estimate the HC of the 4103 household observations of the 1983 U.S. Federal Reserve Board FRB sample survey of consumer finances. Hence, the choice of indicators from this sample survey allows the specification of a multivariate equation to estimate the household HC as a latent variable. 6 . 1 . Estimation of the 1983 U.S. a6erage household HC Using the magnetic tape of the 1983 FRB sample survey and the technical manual and code book prepared by Avery and Elliehausen 1988, we obtain, by age of the household head, i the households earned income; ii the weight each household represents in the U.S. population; and iii the average household earnings. These data are presented in Table 2. Applying to columns 3 and 4 of Table 2 a 3 × 5 weighted moving average, Table 3, columns 2 and 3, presents the smoothed values of the weights and of the average earnings by age of the household head. Fig. 1 presents the observed and the smoothed average household earnings by age x, 20 5 x 5 70.Applying Eq. 13 to column 3 of Table 3 and using the life table from the 1980 US Census, we estimate the average household HC by age of the head, for i = 0.06 and i = 0.08, which are presented in Fig. 2 and Table 3, columns 4 and 5. Since we work with the average earnings by age, yx is the representative earnings corresponding to the 1983 U.S. level of employment of households by age x of the household head. Hence, it corresponds to the term yxEx in Farr, Dublin and Lotka, and in our Eq. 4. Table 2 1983 U.S. household earned income by age of the head Average earnings Earned income Weight Age X 310993520 43911 7082.36 17 7494.07 167202 1253023148 18 317578 1814350558 5713.09 19 508865 4085406294 8028.47 20 8313.55 927902 21 7714158765 11563.34 22 14188201132 1226999 13067.65 1582425 23 20678578739 1601466 27934329403 17442.97 24 14272.22 1522854 21734510178 25 1782115 35785411475 20080.30 26 41106061577 1996990 20584.01 27 20102.15 2127668 42770706090 28 19380.43 29 33607759365 1734108 19708.46 1674046 30 32992876218 1665612 34101152219 20473.65 31 27284.59 1983027 54106076043 32 1642890 52286838631 31826.13 33 1808874 44949950239 24849.69 34 25102.97 1936784 35 48619033230 1720294 55081526927 32018.67 36 26816.11 37 31268954596 1166051 31977.05 1758470 56230680857 38 1379771 45407049871 32909.12 39 1889713 61416993035 32500.70 40 32931.08 1214020 41 39978989053 27006.66 42 34572872918 1280161 36342.18 1366267 43 49653125187 1364019 47005031456 34460.69 44 29801.43 1159862 34565541384 45 1479330 45506665166 30761.67 46 28114.05 47 32839598198 1168085 26724.51 1219288 48 32584876063 39299.87 49 46331162357 1178914 31764.69 1087958 50 34558652908 1295198 36509124986 28188.06 51 29959.03 1266198 37934064576 52 1007671 25427961580 25234.39 53 1062999 40070381284 37695.60 54 26547.38 1262244 33509276024 55 39912.61 56 45584953049 1142119 38051.60 1498842 57 57033329568 732678 16288960432 22232.09 58 25010.57 1324335 33122366845 59 1130870 21994044784 19448.78 60 1446875 38017710278 26275.74 61 20101.49 1407504 62 28292920885 878175 24921129672 28378.32 63 22057.97 64 24847527954 1126465 17564.29 18494999398 65 1052989 Table 2 Continued Average earnings Age X Earned income Weight 1116291 66 9886.66 11036390439 1204940 6276.71 7563060639 67 68 6512401439 1015925 6410.32 9059401881 69 9358.02 968090 70 3625.89 948991 3440934512 873677 9389.24 8203166986 71 72 1227720784 1685.30 728488 2793751660 1008679 73 2769.71 Because the maintenance cost cx is not considered, columns 4 and 5 of Table 3 present the level of gross HC by age at 6 and 8 discount rates, respectively, and because we did not make any assumption of productivity change by age, we call them ‘cross-section’ HC estimates. Therefore, to estimate the average household HC at age x, the average earnings of this representative household, t years later, is assumed to be the average earnings of the households of age x + t. To obtain the life cycle HC estimation of the representative household by age of the head, we apply Eq. 15, where r stands for the annual rate of productivity growth. We assume that r = 0.03 for 20 5 x 5 29, because of the more intensive HC formation in this interval of age; r = 0.02 for 30 5 x 5 54; r = 0.01 for 55 5 x 5 64; and r = 0 for x ] 65. The gross values of the average life cycle HC by age of the household head, for i = 0.06 and 0.08, are presented in Fig. 3, and in Table 3, column 6, and 7. Applying Eq. 14 to columns 4 and 5 of Table 3 and using the weights given in column 2, we obtain the 1983 U.S. average household HC, i.e. the cross-section averages, A6HCh = 283 313, for i = 0.06 and r = 0; 20a A6 HCh = 238 703, for i = 0.08 and r = 0; 20b The 1983 U.S. FRB sample survey gives the following average household wealth Dagum, 1994, A6K = 92 028. Applying Eq. 16 to columns 6 and 7 of Table 3 and using the weights given in column 2, we obtain the estimates of the average life cycle HC of a 20 years old U.S. household head in 1983, i.e. A6HCh = 364 869, for i = 0.06 and r 0; 21a A6HCh = 303 336, fori = 0.08 and r 0. 21b A comparative analysis of these estimated values with those of Jorgenson and Fraumeni, Kendrick, and Machlem is now presented. Table 3 1983 U.S. household human capital by age of the head present values at 6 and 8 discount rates Age Weights 3×5 Average Average HC life cycle Average HC cross section hx; r 0 hx; r = 0 ma earnings 3×5 ma 6 8 6 yx 8 fx X 20 9580.07 356999.53 269275.56 488441.79 358779.74 645184.87 368669.36 280779.37 497328.18 369170.15 21 903998.13 11219.53 379309.79 291441.93 504959.49 12466.29 378514.02 22 1151538.10 389278.39 301622.44 511714.74 23 387185.38 1361677.50 14059.69 398149.86 310894.19 517196.66 15600.93 394781.31 24 1537557 17314.40 1682186.80 405924.33 319248.98 521444.00 401326.15 25 412343.35 326419.31 524239.41 18424.36 406590.37 1778711.90 26 19287.64 1833983.70 417964.13 332961.16 526169.16 411156.20 27 28 423012.43 1845139.10 339100.36 527486.37 415273.88 19721.94 427904.06 345264.47 528618.46 20522.20 419381.46 1846520.80 29 432250.03 351067.04 529203.61 30 423118.21 1805504.50 21757.61 435550.89 356004.78 528562.21 23373.71 425797.04 31 1777239.50 24879.74 1767421.30 437361.63 359614.70 526278.02 426975.28 32 437699.09 361900.95 522390.19 26299.30 426673.04 1793567 33 27318.37 1760263.30 436555.88 362842.17 516915.03 424886.63 34 28085.31 1717149 434298.10 362788.66 510244.38 421984.72 35 431111.22 361920.93 502577.74 28625.95 418156.72 36 1641782.40 29760.09 1617742.80 427204.97 360439.98 494142.85 413615.25 37 421926.38 357672.42 484316.95 38 407711.45 1552246.30 30936.46 415153.07 353474.04 473008.85 31575.03 400328.04 1522963.90 39 31864.70 1470672.80 407357.03 348304.44 460711.75 391942.60 40 41 32217.89 398855.40 342470.61 447761.36 382877.53 1451083.10 389529.39 335843.40 434058.16 32451.88 373020.69 42 1375229.40 379462.76 328494.42 419706.75 43 362462.51 1343209.90 32246.43 369079.66 320843.52 405144.11 32018.38 351634.26 44 1304768.70 31330.15 1305185.70 358425.19 312926.64 390432.06 340587.15 45 347983.07 305231.46 376058.66 31039.57 329812.63 1275575.10 46 30761.43 1248642.50 337357.63 297359.06 361640.80 318924.11 47 30974.24 1219233.10 326521.76 289280.76 347163.71 307905.62 48 314925.62 280436.10 332100.90 31028.38 296221.18 49 1208704.90 30926.05 1188862.50 302668.06 270915.10 316574.02 283985.56 50 289909.34 260864.36 300768.25 51 271373.12 1173567.90 30856.28 276591.13 250213.36 284651.14 30298.21 258343.45 1163351.50 52 30565.28 1157037.70 263200.54 239446.16 268727.77 245403.98 53 248834.08 54 227641.22 1173961.10 252131.55 231676.99 31665.07 232526.19 213793.82 233956.41 33113.58 216226.54 1160932.50 55 213744.20 197319.61 213399.53 56 198204.34 1175531.70 32873.16 194106.06 179809.43 192090.34 31567.43 179214.30 57 1165862.90 29349.38 1194844.10 174633.73 162278.39 171053.03 160275.53 58 156273.58 145681.50 151227.66 26559.50 142338.60 1200313.70 59 24533.32 1224241.40 139767.80 130776.55 133339.93 126147.55 60 1214660.70 23169.02 124331.85 116793.61 116600.10 110928.13 61 109314.88 103078.04 100355.87 23162.18 96026.093 62 1205977.10 21748.76 1165554.70 93234.536 88116.812 83139.883 79945.357 63 77481.446 64 73291.741 1124819.50 66345.12 64072.806 19501.23 62969.106 59521.144 50874.56 15956.31 49320.06 1098459.60 65 Table 3 Continued Weights 3×5 Average HC life cycle Average Age Average HC cross section hx; r 0 ma earnings 3×5 hx; r = 0 ma 8 6 yx 8 6 X fx 12796.08 51193.525 48334.018 38023.44 37016.188 66 1083580.30 41916.582 39526.982 27539.455 67 1075272.10 9841.40 26938.753 19374.914 33111.415 35114.252 19070.453 8004.14 68 1041606.30 6812.69 29765.067 28086.203 12484.341 12379.31 69 986735.40 6248.6197 23879.844 6248.6197 70 938314.60 6248.62 25287.276 Fig. 1. 1983 U.S. observed and smoothed 3 × 5 ma household earned income by age of the head. Fig. 2. 1983 U.S. household human capital by age of the head cross-section at 6 and 8 discount rates. Fig. 3. 1983 U.S. household human capital by age of the head with productivity increase life cycle at 6 and 8 discount rates. Jorgenson and Fraumeni’s full market and non-market human and non-human wealth estimates are incommensurable with ours because of these authors’ inclusion of the non-market accounts with the scope of providing ‘‘a comprehensive perspec- tive on the role of capital formation in U.S. economic growth’’ Jorgenson and Fraumeni, 1989. The incommensurability is strongly influenced by the authors’ very high values of the non-market labor income estimations, which are almost five times the market labor income estimations see Jorgenson and Fraumeni, 1989 Tables 5.13 and 5.14, and the authors’ full labor income estimations for the period 1948 – 1984, which are between nine and 14 times the full property income see Jorgenson and Fraumeni, 1989 Tables 5.15 and 5.16. This large fluctuation in the full laborfull property income ratio should be imputable to the method of estimating the households non-market labor outlays and labor incomes. Should this be the case, the assumptions leading to these estimations would require a thorough reassessment. Jorgenson and Fraumeni provided a valuable and original contribution by incorporating the non-market activities and offering an ingenious approach to their estimations. Our observations and comments purport to point out the need of reassessing and refining their approach. Our observations are further supported by the following 1982 household averages obtained from Jorgenson and Fraumeni’s Table 5.32, for a population of household equal to 83 918 000, 1. the 1982 U.S. average of the full market and non-market household non-hu- man wealth is equal to 165 494; 2. the 1982 U.S. average of the full market and non-market household human wealth is equal to 1 989 924. Therefore, Jorgenson and Fraumeni’s full non-human wealth estimation in 1982 is 80 higher than the 1982 average household non-human wealth of 92 028 given by the 1983 U.S. FRB sample survey. On the other hand, Jorgenson and Frau- meni’s full human wealth estimation in 1982 is 556 i.e. 6.56 times higher than our 1982 average household market human wealth. Can this huge difference be imputable to the non-market human wealth? Jorgenson and Fraumeni’s contribu- tion does not provide enough information to answer this question. Besides, their full average household human wealth estimations for the period 1948 – 1984 are so high that, for a moderate level of human wealth inequality and rate of return, it would imply the practical inexistence of poverty or a very high percentage of HC idle capacity. In effect, at a reasonable 8 rate of return, the 1982 average full household human wealth of 1 989 924 would give an average household earned income of almost 160 000 The estimations of Kendrick, 1976 regarding non-human and human wealth end in 1969. Using Jorgenson and Fraumeni’s rate of growth Kendrick’s 1969 estimations of the non-human and human wealth extrapolated to 1982 give 77 544 and 135 883, respectively. These estimations represent an 84.3 of the observed 1983 U.S. FRB average household non-human wealth, and a 44.8 of our life cyle average household human wealth estimation, using the 8 discount rate. Macklem’s 1982 estimations for Canada, are, i.a per capita non-human wealth in Canadian dollars of 1986 is equal to Can40 906; ii.a per capita human wealth in Canadian dollars of 1986 is equal to Can238 043. For a 3.2 average Canadian household size, we have, i.b average household non-human wealth in Canadian dollars of 1986 is equal to Can130 899; ii.b average household human wealth in Canadian dollars of 1986 is equal to Can761 738. Taking into account the Canadian’s rate of inflation from 1982 to 1986 and the 1982 Canadian – U.S. exchange rate, we obtain an average household non-human wealth marginally higher than the U.S. while the Canadian average household human wealth is about twice our estimate for the US. There is not a good reason to account for so large difference. Besides, from 1963 to 1994, Macklem’s estima- tions present large, unacceptable and unsubstantiated fluctuations. For instance, from 1963 to 1975, the human wealth estimations show an increasing trend followed by a decreasing trend until 1984, then an increasing one until 1988, then decreasing until 1990, followed by an increasing trend. Besides, the maximum human wealth estimation is obtained in 1988, which is very close to the 1975 estimation. These unbecoming fluctuations for an economy presenting a steady economic growth trend should be imputed to the limitations of the exogenous variables specified in the bivariate autoregressive model. 6 . 2 . Estimation of HC of each sample obser6ation and its size distribution From the 1983 U.S. sample survey, we select the following indicators to estimate the latent HC variable by household: x 1 , age of the household head; x 2 , region of residence; x 3 , marital status; x 4 , sex of the spouse; x 5 , years of schooling of the household head; x 6 , years of schooling of the spouse; x 7 , number of children; x 8 , years of full-time work of the household head; x 9 , years of full-time work of the spouse; x 10 = k, total wealth; and x 11 = u , total debt. Hence, the latent variable z in Eq. 10 is specified as a linear function of qualitative and quantitative variables. Applying an iterative algorithm to estimate a latent variable as in Wold 1982 partial least squares and Dagum and Vittadini 1996, further developed by several authors to deal with the specification of qualitative and quantitative variables as in Young et al. 1976, Haagen and Vittadini 1998, Vittadini 1999, we obtain the following estimation of z i , i = 1, 2, …, 4103, where the numbers in parentheses are the Student-t: z i = − 0.222x 1i − − 1.3 0.267x 2i − 25.1 + 0.115x 3i 6.0 − 0.087x 4i − 5.2 + 0.334x 5i 31.9 + 0.570x 6i 29.5 + 0.045x 7i 3.6 + 0.042x 8i 2.6 − 0.088x 9i − 7.6 + 0.090x 10i 8.3 + 0.154x 11i ; 14.7 22 where z i and x ij , j = 1, 2, …, 11, are standardized variables. The corresponding R 2 and F values are, R 2 = 0.618 and F11, 4091 = 602.9. Given that we are working with a cross-section sample of 4103 observations, the coefficient of determination R 2 and the F value are exceptionally high, hence, clearly accepting the goodness of fit of Eq. 22 even at the 1 level of significance. In effect, at the 5 and1 significance levels, the F critical values with 11 and 4091 degrees of freedom are 1.31 and 2.30, respectively. Applying the transformation Eq. 11 to the estimations of z i in Eq. 22, we obtain the accounting monetary estimations of h i , i = 1, 2, …, 4103, each one having a weight that corresponds to the number of households each sample observation represents in the 1983 U.S. population. Dividing each h i by its mean obtained from Eq. 12 and multiplying by the average household HC given by Eq. 20a or Eq. 20b, we obtain the household HC distribution in US dollars, with mean equal to 283 313, if the flow of average earnings is actualized at 6, or with mean equal to 238 703, if it is actualized at 8 discount rate. 6 . 3 . Fitting the household HC distribution Applying the non-linear least squares method of parameters estimation, the sample estimation data of the household HC distribution at 8 discount rate are fitted to Dagum three- and four-parameter model given in Eqs. 18 and 19. The fitted three-parameter model Eq. 18 is, Fh = 1 + 3146.48h − 2.3421 − 0.4101 , h \ 0, b, l \ 0, SSE = 0.00199, K-S = 0.039, 23 whereas for the four-parameter model Eq. 19, we have, Fh = − 0.0762 + 1.07621 + 3134.81h − 2.2929 − 0.341 , h \ h ] 0, SSE = 0.00075, K-S = 0.012, h = 1.1326. 24 where h is measured in 10 000, SSE stands for the sum of the squares errors, K – S for the Kolmogorov – Smirnov statistic, and h is the solution of Fh = 0 in Eq. 24, i.e. h = l 1d 1 − 1 a 1b − 1 n − 1d = 1.1326. 25 The HC histogram and its fitted probability density function corresponding to the estimated four-parameter Dagum model Eq. 24 are presented in Fig. 4, which shows further evidences of its excellent goodness of fit. The asymptotic critical value of the K – S statistic at 10, 5, and 1 significance levels are K – S0.10 = 1.224103 12 = 0.019, K – S0.05 = 1.364103 12 = 0.021, and K – S0.01 = 1.634103 12 = 0.025, respectively. Hence, according to this approxi- mated criterion to test the goodness of fit, the four-parameter fitted model clearly accepts it whereas the three-parameter does not. We call it approximated criterion or a proxy test because we are working with the fitted model instead of an independently specified distribution, hence the significance level should be larger than in the case of two independently observed or specified distributions. The goodness of fit of the four-parameter model continues to be accepted even at the 10 significance level because of its very low K – S statistic. Fig. 4. 1983 U.S. human capital distribution: histogram and Dagum fitted model. A way to test if there is a statistically significant reduction in the SSE, when passing from the three- to the four-parameter model, makes use of the F-test. Since we have one extra parameter and we fit both models to the HC data grouped in 47 class intervals, we have 42 degrees of freedom for the four-parameter model. Therefore, F1, 42 = 0.00199 − 0.00075 0.0007542 = 69.5, whereas the critical level at 5 significant level is F 0.05 1, 42 = 4.07. Hence, the null hypothesis that the four-parameter model does not significantly reduce the SSE of the three-parameter fitted model is strongly rejected. Since the estimate of a is significantly different from zero, and is negative, the estimated model Eq. 24 is Dagum type III. Translating the ordinate to the point h = 1.1326, we have, Fh = [1 + 3134.81h − 1.1326 − 2.2929 ] − 0.341 , h \ 1.1326. 26 The solution Fh = 0.5 in Eq. 24 is the estimated median of the fitted distribution. It is equal to 163 539 whereas the median of the sample values of h is equal to 163 061, i.e. the model estimation and the sample value of the median are almost identical, further supporting the goodness of fit of Eq. 24. The mathematical expectation the estimated mean is equal to 255 367 and the sample mean is equal to 238 703, presenting a discrepancy of less than 7. The Gini ratio is G = 0.528, which is smaller than the Gini ratio for the total wealth G = 0.636 and for the net wealth G = 0.681, and greater than the income inequality G = 0.444, which are estimated from the same household sample survey Dagum, 1994.

7. Conclusion