The Human Development Index as Isoelastic GDP : Evidence from China and Pakistan

Gross domestic product (GDP) is shown to possess three new desiderata. First, GDP is almost perfectly correlated over time with the first principal component of its three classical indicators. Second, this principal component is in a class of weighted indexes ancillary to GDP. Each ancillary index informs policy as to allocation of resources over the three GDP indicators. Third, a country-specific power of GDP almost perfectly predicts the United Nation’s Human Development Index (HDI). These findings are brought by principal components and regression analyses of time series supplied by the World Bank and the United Nations. Axiomatic HDI computation is carried out without survey sampling, probabilistic inference, significance testing, or even HDI data.


The Keynesian Construct
This paper views GDP's three classic constituents as separate time-varying indicators.Household expenditure and domestic savings were introduced as elements of GDP during the Great Depression in 1936 by John Maynard Keynes in his General Theory of Employment, Interest and Money (Keynes 1936).In 1940, Keynes added government expenditure to GDP in How to Pay for the War (Keynes 1940)."Shortly before his death on 21 April 1946, Keynes persuaded the powers at the University of Cambridge to create a new Department of Applied Economics.[ . . . ] the Cambridge department along with Harvard University's Development Advisory Service would together [ . . . ] incubate the first set of ideas around what GDP would look like, and then help to export them to the four corners of the world."(Masood 2016, p. 32).American planners then used the Keynesian GDP formula to measure the effect of American aid and to manage European economies.In 1999, overlooking the fact that GDP was "Made in Cambridge" (Masood 2016, pp. 31-37), the United States Commerce Department proclaimed the GDP formula as the US government's greatest invention of the 20th century.
The severest criticism of GDP has been that it does not measure well-being." [ . . . ] after the end of World War II, [ . . .] Amartya Sen and Mahbub ul Haq would openly revolt against the idea of organizing economies according to GDP. And Haq [ . . . ] would lead the design of the United Nations' Human Development Index, which has so far come closest to dethroning GDP" (Masood 2016, p. 41).In 1989, Haq's UN team settled on life expectancy, education, and per capita income as the components of the HDI.The last component was insisted on by the "formidable Sen", who resolved the measurement of life expectancy and education in years and income in dollars (Masood 2016, pp. 93-95).However, "The HDI, for all its successes, had no discernable impact on the dominance of GDP as the world's principal and most sought-after measure of economic well-being" (Masood 2016, p. 101).
government expenditures for purchases of goods and services (including compensation of employees).It also includes most expenditures on national defense and security, but excludes government military expenditures that are part of government capital formation.Data are in current U.S. dollars.

HDI
The HDI comprises macro indicators that are described by the United Nations Development Program (http://hdr.undp.org/en/data): The HDI is a summary measure of average achievement in key dimensions of human development: a long and healthy life, being knowledgeable, and having a decent standard of living.The HDI is the geometric mean of normalized indices for each of the three dimensions.
The health dimension is assessed by life expectancy at birth, the education dimension is measured by mean of years of schooling for adults aged 25 years and more and expected years of schooling for children of school entering age.The standard of living dimension is measured by gross national income per capita.The HDI uses the logarithm of income to reflect the diminishing importance of income with increasing GNI.The scores for the three HDI dimension indices are then aggregated into a composite index using their geometric mean.
The normalized [0, 1] scale for health and education (in years) and standard of living (in logarithm-of-dollar-units) is obtained as follows: Minimum and maximum values (goalposts) are set in order to transform the indicators expressed in different units into indices on a scale of 0 to 1.These goalposts act as the "natural zeros" and "aspirational targets," respectively, from which component indicators are standardized.Having defined the minimum and maximum values, each dimension index is calculated as the ratio of actual value less minimum value to maximum value less minimum value.
For the education dimension, this ratio is first applied to each of the two indicators, and then the arithmetic mean of the two resulting indices is taken.Because each dimension index is a proxy for capabilities in the corresponding dimension, the transformation function from income to capabilities is likely to be concave-that is, each additional dollar of income has a smaller effect on expanding capabilities.Thus, for income, the natural logarithm of the actual, minimum, and maximum values is used.

Indicator Weighting
The theory here postulates latent population distributions akin to those in Coombs' theory of psychological data (Coombs 1950(Coombs , 1964)).Each of the three GDP indicators in Section 2 is posited to take an unobservable distribution over individual-time points in a particular nation.Internally consistent GDP indexes are then constructed for China and Pakistan from a latent 2-level principal-components analysis implied by axioms 1 and 2 in Section 4.2.This approach derives optimal indicator weights from the postulated individual-time points in each country.It differs from standard GDP calculation, which equally weights the indicators in Section 2 by an arithmetic summation.It also differs from other indexes, which weight their indicators to maximize the prediction of external criteria.

Latent 2-Level Principal Components Analysis
This section treats latent individual-time points on the real line, with individuals nested within successive years for a given nation (Bechtel 2017;Johnston 1984, pp. 536-44).
Existence Axiom 1.A nation's GDP indicator G tj (j = 1, 2, 3) in year t is the mean

HDI
The HDI comprises macro indicators that are described by the United Nations Development Program (http://hdr.undp.org/en/data): The HDI is a summary measure of average achievement in key dimensions of human development: a long and healthy life, being knowledgeable, and having a decent standard of living.The HDI is the geometric mean of normalized indices for each of the three dimensions.
The health dimension is assessed by life expectancy at birth, the education dimension is measured by mean of years of schooling for adults aged 25 years and more and expected years of schooling for children of school entering age.The standard of living dimension is measured by gross national income per capita.The HDI uses the logarithm of income to reflect the diminishing importance of income with increasing GNI.The scores for the three HDI dimension indices are then aggregated into a composite index using their geometric mean.
The normalized [0, 1] scale for health and education (in years) and standard of living (in logarithm-of-dollar-units) is obtained as follows: Minimum and maximum values (goalposts) are set in order to transform the indicators expressed in different units into indices on a scale of 0 to 1.These goalposts act as the "natural zeros" and "aspirational targets," respectively, from which component indicators are standardized.Having defined the minimum and maximum values, each dimension index is calculated as the ratio of actual value less minimum value to maximum value less minimum value.
For the education dimension, this ratio is first applied to each of the two indicators, and then the arithmetic mean of the two resulting indices is taken.Because each dimension index is a proxy for capabilities in the corresponding dimension, the transformation function from income to capabilities is likely to be concave-that is, each additional dollar of income has a smaller effect on expanding capabilities.Thus, for income, the natural logarithm of the actual, minimum, and maximum values is used.

Indicator Weighting
The theory here postulates latent population distributions akin to those in Coombs' theory of psychological data (Coombs 1950(Coombs , 1964)).Each of the three GDP indicators in Section 2 is posited to take an unobservable distribution over individual-time points in a particular nation.Internally consistent GDP indexes are then constructed for China and Pakistan from a latent 2-level principalcomponents analysis implied by axioms 1 and 2 in Section 4.2.This approach derives optimal indicator weights from the postulated individual-time points in each country.It differs from standard GDP calculation, which equally weights the indicators in Section 2 by an arithmetic summation.It also differs from other indexes, which weight their indicators to maximize the prediction of external criteria.

Latent 2-Level Principal Components Analysis
This section treats latent individual-time points on the real line, with individuals nested within successive years for a given nation (Bechtel 2017;Johnston 1984, pp. 536-44).Gtj (j = 1,2,3) in year t is the mean G t.j of Nt dollar denominated impacts Gtij on individuals i = 1,…, Nt, where Nt is population size in t = 1990,…, 2015.osts) are set in order to transform the indicators le of 0 to 1.These goalposts act as the "natural zeros" which component indicators are standardized.alues, each dimension index is calculated as the ratio lue less minimum value.rst applied to each of the two indicators, and then the taken.Because each dimension index is a proxy for transformation function from income to capabilities l dollar of income has a smaller effect on expanding ithm of the actual, minimum, and maximum values ion distributions akin to those in Coombs' theory of f the three GDP indicators in Section 2 is posited to idual-time points in a particular nation.Internally r China and Pakistan from a latent 2-level principald 2 in Section 4.2.This approach derives optimal l-time points in each country.It differs from standard dicators in Section 2 by an arithmetic summation.It eir indicators to maximize the prediction of external oints on the real line, with individuals nested within 7; Johnston 1984, pp. 536-44).

Existence Axiom 1. A nation's GDP indicator
tj (j = 1, 2, 3)  education (in years) and standard of living (in s) are set in order to transform the indicators of 0 to 1.These goalposts act as the "natural zeros" which component indicators are standardized.es, each dimension index is calculated as the ratio e less minimum value.applied to each of the two indicators, and then the ken.Because each dimension index is a proxy for ransformation function from income to capabilities ollar of income has a smaller effect on expanding m of the actual, minimum, and maximum values distributions akin to those in Coombs' theory of the three GDP indicators in Section 2 is posited to al-time points in a particular nation.Internally hina and Pakistan from a latent 2-level principal-2 in Section 4.2.This approach derives optimal ime points in each country.It differs from standard cators in Section 2 by an arithmetic summation.It r indicators to maximize the prediction of external nts on the real line, with individuals nested within Johnston 1984, pp. 536-44).
(j = 1, 2, 3) in year t is the mean G t.j of Nt dollar here Nt is population size in t = 1990,…, 2015.tij constitute ∑tNt latent individual-time points.
Homogeneity Axiom 2. The 3 × 3 within-year covariance matrix of the vector ( also differs from other indexes, which weight their indicators to maximize the prediction of external criteria.

Latent 2-Level Principal Components Analysis
This section treats latent individual-time points on the real line, with individuals nested within successive years for a given nation (Bechtel 2017;Johnston 1984, pp. 536-44).Gtj (j = 1,2,3) in year t is the mean G t.j of Nt dollar denominated impacts Gtij on i = 1,…, Nt, where Nt is population size in t = 1990,…, 2015.ti1 also differs from other indexes, which weight their indicators to maximize the prediction of external criteria.

Latent 2-Level Principal Components Analysis
This section treats latent individual-time points on the real line, with individuals nested within successive years for a given nation (Bechtel 2017;Johnston 1984, pp. 536-44).

Existence Axiom 1. A nation's GDP indicator Gtj
(j = 1, 2, 3) in year t is the mean G t.j of Nt dollar denominated impacts Gtij on individuals i = 1,…, Nt, where Nt is population size in t = 1990,…, 2015.ti2 also differs from other indexes, which weight their indicators to maximize the prediction of external criteria.

Latent 2-Level Principal Components Analysis
This section treats latent individual-time points on the real line, with individuals nested within successive years for a given nation (Bechtel 2017;Johnston 1984, pp. 536-44).
Corollary 1.The covariance matrix of the vector ( alysis implied by axioms 1 and 2 in Section 4.2.This approach derives optimal ts from the postulated individual-time points in each country.It differs from standard , which equally weights the indicators in Section 2 by an arithmetic summation.It other indexes, which weight their indicators to maximize the prediction of external el Principal Components Analysis treats latent individual-time points on the real line, with individuals nested within for a given nation (Bechtel 2017;Johnston 1984, pp. 536-44).
m 1.A nation's GDP indicator Gtj (j = 1, 2, 3) in year t is the mean G t.j of Nt dollar acts Gtij on individuals i = 1,…, Nt, where Nt is population size in t = 1990,…, 2015. ti1 ts analysis implied by axioms 1 and 2 in Section 4.2.This approach derives optimal eights from the postulated individual-time points in each country.It differs from standard lation, which equally weights the indicators in Section 2 by an arithmetic summation.It from other indexes, which weight their indicators to maximize the prediction of external -Level Principal Components Analysis ection treats latent individual-time points on the real line, with individuals nested within years for a given nation (Bechtel 2017;Johnston 1984, pp. 536-44).
Axiom 1.A nation's GDP indicator Gtj (j = 1, 2, 3) in year t is the mean G t.j of Nt dollar d impacts Gtij on individuals i = 1,…, Nt, where Nt is population size in t = 1990,…, 2015. ti2 onents analysis implied by axioms 1 and 2 in Section 4.2.This approach derives optimal tor weights from the postulated individual-time points in each country.It differs from standard calculation, which equally weights the indicators in Section 2 by an arithmetic summation.It iffers from other indexes, which weight their indicators to maximize the prediction of external a.
tent 2-Level Principal Components Analysis his section treats latent individual-time points on the real line, with individuals nested within sive years for a given nation (Bechtel 2017;Johnston 1984, pp. 536-44).
Corollary 2. Individual ti's latent score on the first principal component psychological data (Coombs 1950(Coombs , 1964)).Each of the three GDP indicators in Section 2 is posited to take an unobservable distribution over individual-time points in a particular nation.Internally consistent GDP indexes are then constructed for China and Pakistan from a latent 2-level principalcomponents analysis implied by axioms 1 and 2 in Section 4.2.This approach derives optimal indicator weights from the postulated individual-time points in each country.It differs from standard GDP calculation, which equally weights the indicators in Section 2 by an arithmetic summation.It also differs from other indexes, which weight their indicators to maximize the prediction of external criteria.

Latent 2-Level Principal Components Analysis
This section treats latent individual-time points on the real line, with individuals nested within successive years for a given nation (Bechtel 2017;Johnston 1984, pp. 536-44).Gtj   (j = 1, 2, 3) in year t is the mean G t.j of Nt dollar denominated impacts Gtij on individuals i = 1,…, Nt, where Nt is population size in t = 1990,…, 2015. of variables psychological data (Coombs 1950(Coombs , 1964)).Each of the three GDP indicators in Section 2 is posited to take an unobservable distribution over individual-time points in a particular nation.Internally consistent GDP indexes are then constructed for China and Pakistan from a latent 2-level principalcomponents analysis implied by axioms 1 and 2 in Section 4.2.This approach derives optimal indicator weights from the postulated individual-time points in each country.It differs from standard GDP calculation, which equally weights the indicators in Section 2 by an arithmetic summation.It also differs from other indexes, which weight their indicators to maximize the prediction of external criteria.

Latent 2-Level Principal Components Analysis
This section treats latent individual-time points on the real line, with individuals nested within successive years for a given nation (Bechtel 2017;Johnston 1984, pp. 536-44).Gtj   (j = 1, 2, 3) in year t is the mean G t.j of Nt dollar denominated impacts Gtij on individuals i = 1,…, Nt, where Nt is population size in t = 1990,…, 2015.ti1 , psychological data (Coombs 1950(Coombs , 1964)).Each of the three GDP indicators in Section 2 is posited to take an unobservable distribution over individual-time points in a particular nation.Internally consistent GDP indexes are then constructed for China and Pakistan from a latent 2-level principalcomponents analysis implied by axioms 1 and 2 in Section 4.2.This approach derives optimal indicator weights from the postulated individual-time points in each country.It differs from standard GDP calculation, which equally weights the indicators in Section 2 by an arithmetic summation.It also differs from other indexes, which weight their indicators to maximize the prediction of external criteria.

Latent 2-Level Principal Components Analysis
This section treats latent individual-time points on the real line, with individuals nested within successive years for a given nation (Bechtel 2017;Johnston 1984, pp. 536-44).Gtj   (j = 1, 2, 3) in year t is the mean G t.j of Nt dollar denominated impacts Gtij on individuals i = 1,…, Nt, where Nt is population size in t = 1990,…, 2015.ti2 , and psychological data (Coombs 1950(Coombs , 1964)).Each of the three GDP indicators in Section 2 is posited to take an unobservable distribution over individual-time points in a particular nation.Internally consistent GDP indexes are then constructed for China and Pakistan from a latent 2-level principalcomponents analysis implied by axioms 1 and 2 in Section 4.2.This approach derives optimal indicator weights from the postulated individual-time points in each country.It differs from standard GDP calculation, which equally weights the indicators in Section 2 by an arithmetic summation.It also differs from other indexes, which weight their indicators to maximize the prediction of external criteria.

Latent 2-Level Principal Components Analysis
This section treats latent individual-time points on the real line, with individuals nested within successive years for a given nation (Bechtel 2017;Johnston 1984, pp. 536-44).Gtj   (j = 1, 2, 3 ulation distributions akin to those in Coombs' theory of ach of the three GDP indicators in Section 2 is posited to ndividual-time points in a particular nation.Internally ed for China and Pakistan from a latent 2-level principal-1 and 2 in Section 4.2.This approach derives optimal idual-time points in each country.It differs from standard e indicators in Section 2 by an arithmetic summation.It ht their indicators to maximize the prediction of external lysis me points on the real line, with individuals nested within l 2017; Johnston 1984, pp. 536-44).

Existence Axiom 1. A nation's GDP indicator
ator Gtj (j = 1, 2, 3) in year t is the mean G t.j of Nt dollar , Nt, where Nt is population size in t = 1990,…, 2015.ti3 , where the vector (a 1 a 2 a 3 ) is the first eigenvector of the covariance matrix of ( in to those in Coombs' theory of dicators in Section 2 is posited to n a particular nation.Internally an from a latent 2-level principal-.This approach derives optimal h country.ti3 ) (Bechtel 2017;Johnston 1984, pp. 536-44).
Lemma 1.If a 1 2 + a 2 2 + a 3 2 = 1, then the variance of Theory hting ere postulates latent population distributions akin to those in Coombs' theory of a (Coombs 1950(Coombs , 1964)).Each of the three GDP indicators in Section 2 is posited to able distribution over individual-time points in a particular nation.Internally dexes are then constructed for China and Pakistan from a latent 2-level principalysis implied by axioms 1 and 2 in Section 4.2.This approach derives optimal from the postulated individual-time points in each country.It differs from standard which equally weights the indicators in Section 2 by an arithmetic summation.It ther indexes, which weight their indicators to maximize the prediction of external

Principal Components Analysis
reats latent individual-time points on the real line, with individuals nested within or a given nation (Bechtel 2017;Johnston 1984, pp. 536-44).

A nation's GDP indicator Gtj
(j = 1, 2, 3) in year t is the mean G t.j of Nt dollar ts Gtij on individuals i = 1,…, Nt, where Nt is population size in t = 1990,…, 2015.over i = 1, . . ., N t and t = 1990, . . ., 2015 equals the first eigenvalue λ of the covariance matrix of ( tulates latent population distributions akin to those in Coombs' theory of bs 1950Coombs' theory of bs , 1964)).Each of the three GDP indicators in Section 2 is posited to stribution over individual-time points in a particular nation.Internally re then constructed for China and Pakistan from a latent 2-level principalplied by axioms 1 and 2 in Section 4.2.This approach derives optimal e postulated individual-time points in each country.It differs from standard qually weights the indicators in Section 2 by an arithmetic summation.It exes, which weight their indicators to maximize the prediction of external l Components Analysis tent individual-time points on the real line, with individuals nested within en nation (Bechtel 2017;Johnston 1984, pp. 536-44).given nation (Bechtel 2017;Johnston 1984, pp. 536-44).
A nation's GDP indicator Gtj (j = 1, 2, 3) in year t is the mean G t.j of Nt dollar tij on individuals i = 1,…, Nt, where Nt is population size in t = 1990,…, 2015.
ti2 ta Theory ghting here postulates latent population distributions akin to those in Coombs' theory of ta (Coombs 1950(Coombs , 1964)).Each of the three GDP indicators in Section 2 is posited to vable distribution over individual-time points in a particular nation.Internally indexes are then constructed for China and Pakistan from a latent 2-level principallysis implied by axioms 1 and 2 in Section 4.2.This approach derives optimal s from the postulated individual-time points in each country.It differs from standard , which equally weights the indicators in Section 2 by an arithmetic summation.It other indexes, which weight their indicators to maximize the prediction of external l Principal Components Analysis treats latent individual-time points on the real line, with individuals nested within for a given nation (Bechtel 2017;Johnston 1984, pp. 536-44).indicator Gtj (j = 1, 2, 3) in year t is the mean G t.j of Nt dollar i = 1,…, Nt, where Nt is population size in t = 1990,…, 2015.b , with variance λ(1 + ω) −1 , replicates the yearly mean capabilities.Thus, for income, the natural logarithm of the actual, minimum, and maximum values is used.

Indicator Weighting
The theory here postulates latent population distributions akin to those in Coombs' theory of psychological data (Coombs 1950(Coombs , 1964)).Each of the three GDP indicators in Section 2 is posited to take an unobservable distribution over individual-time points in a particular nation.Internally consistent GDP indexes are then constructed for China and Pakistan from a latent 2-level principalcomponents analysis implied by axioms 1 and 2 in Section 4.2.This approach derives optimal indicator weights from the postulated individual-time points in each country.It differs from standard GDP calculation, which equally weights the indicators in Section 2 by an arithmetic summation.It also differs from other indexes, which weight their indicators to maximize the prediction of external criteria.

Latent 2-Level Principal Components Analysis
This section treats latent individual-time points on the real line, with individuals nested within successive years for a given nation (Bechtel 2017;Johnston 1984, pp. 536-44).
t. = ∑ i capabilities.Thus, for income, the natural logarithm of the actual, minimum, and maximum values is used.

Indicator Weighting
The theory here postulates latent population distributions akin to those in Coombs' theory of psychological data (Coombs 1950(Coombs , 1964)).Each of the three GDP indicators in Section 2 is posited to take an unobservable distribution over individual-time points in a particular nation.Internally consistent GDP indexes are then constructed for China and Pakistan from a latent 2-level principalcomponents analysis implied by axioms 1 and 2 in Section 4.2.This approach derives optimal indicator weights from the postulated individual-time points in each country.It differs from standard GDP calculation, which equally weights the indicators in Section 2 by an arithmetic summation.It also differs from other indexes, which weight their indicators to maximize the prediction of external criteria.

Latent 2-Level Principal Components Analysis
This section treats latent individual-time points on the real line, with individuals nested within successive years for a given nation (Bechtel 2017;Johnston 1984, pp. 536-44).dollar of income has a smaller effect on expanding thm of the actual, minimum, and maximum values n distributions akin to those in Coombs' theory of the three GDP indicators in Section 2 is posited to ual-time points in a particular nation.Internally China and Pakistan from a latent 2-level principal-2 in Section 4.2.This approach derives optimal -time points in each country.It differs from standard icators in Section 2 by an arithmetic summation.It ir indicators to maximize the prediction of external ints on the real line, with individuals nested within ; Johnston 1984, pp. 536-44).tj (j = 1, 2, 3) in year t is the mean G t.j of Nt dollar where Nt is population size in t = 1990,…, 2015.
The within-year vector dices is taken.Because each dimension index is a proxy for sion, the transformation function from income to capabilities ditional dollar of income has a smaller effect on expanding al logarithm of the actual, minimum, and maximum values opulation distributions akin to those in Coombs' theory of ).Each of the three GDP indicators in Section 2 is posited to r individual-time points in a particular nation.Internally ucted for China and Pakistan from a latent 2-level principals 1 and 2 in Section 4.2.This approach derives optimal dividual-time points in each country.It differs from standard s the indicators in Section 2 by an arithmetic summation.It eight their indicators to maximize the prediction of external nalysis l-time points on the real line, with individuals nested within tel 2017; Johnston 1984, pp. 536-44).

−
esulting indices is taken.Because each dimension index is a proxy for ing dimension, the transformation function from income to capabilities is, each additional dollar of income has a smaller effect on expanding , the natural logarithm of the actual, minimum, and maximum values tes latent population distributions akin to those in Coombs' theory of 950, 1964).Each of the three GDP indicators in Section 2 is posited to ution over individual-time points in a particular nation.Internally en constructed for China and Pakistan from a latent 2-level principalby axioms 1 and 2 in Section 4.2.This approach derives optimal stulated individual-time points in each country.It differs from standard lly weights the indicators in Section 2 by an arithmetic summation.It s, which weight their indicators to maximize the prediction of external mponents Analysis individual-time points on the real line, with individuals nested within ation (Bechtel 2017;Johnston 1984, pp. 536-44).
's GDP indicator Gtj (j = 1,2,3)  ) in year t is the mean G t.j of Nt dollar pulation size in t = 1990,…, 2015.t .for i = 1, . . ., N t over t = 1990, . . ., 2015.However, the between-year vector For the education dimension, this ratio is first applied to each of the two indicators, and then the arithmetic mean of the two resulting indices is taken.Because each dimension index is a proxy for capabilities in the corresponding dimension, the transformation function from income to capabilities is likely to be concave-that is, each additional dollar of income has a smaller effect on expanding capabilities.Thus, for income, the natural logarithm of the actual, minimum, and maximum values is used.

Indicator Weighting
The theory here postulates latent population distributions akin to those in Coombs' theory of psychological data (Coombs 1950(Coombs , 1964)).Each of the three GDP indicators in Section 2 is posited to take an unobservable distribution over individual-time points in a particular nation.Internally consistent GDP indexes are then constructed for China and Pakistan from a latent 2-level principalcomponents analysis implied by axioms 1 and 2 in Section 4.2.This approach derives optimal indicator weights from the postulated individual-time points in each country.It differs from standard GDP calculation, which equally weights the indicators in Section 2 by an arithmetic summation.It also differs from other indexes, which weight their indicators to maximize the prediction of external criteria.

Latent 2-Level Principal Components Analysis
This section treats latent individual-time points on the real line, with individuals nested within successive years for a given nation (Bechtel 2017;Johnston 1984, pp. 536-44).

Existence Axiom 1. A nation's GDP indicator Gtj
(j = 1, 2, 3) in year t is the mean G t.j of Nt dollar denominated impacts Gtij on individuals i = 1,…, Nt, where Nt is population size in t = 1990,…, 2015.b , which replicates the mean lue less minimum value.rst applied to each of the two indicators, and then the taken.Because each dimension index is a proxy for transformation function from income to capabilities l dollar of income has a smaller effect on expanding ithm of the actual, minimum, and maximum values ion distributions akin to those in Coombs' theory of f the three GDP indicators in Section 2 is posited to idual-time points in a particular nation.Internally r China and Pakistan from a latent 2-level principald 2 in Section 4.2.This approach derives optimal l-time points in each country.It differs from standard dicators in Section 2 by an arithmetic summation.It eir indicators to maximize the prediction of external oints on the real line, with individuals nested within 7; Johnston 1984, pp. 536-44).ess minimum value to maximum value less minimum value.ucation dimension, this ratio is first applied to each of the two indicators, and then the n of the two resulting indices is taken.Because each dimension index is a proxy for he corresponding dimension, the transformation function from income to capabilities oncave-that is, each additional dollar of income has a smaller effect on expanding us, for income, the natural logarithm of the actual, minimum, and maximum values ata Theory eighting y here postulates latent population distributions akin to those in Coombs' theory of ata (Coombs 1950(Coombs , 1964)).Each of the three GDP indicators in Section 2 is posited to ervable distribution over individual-time points in a particular nation.Internally indexes are then constructed for China and Pakistan from a latent 2-level principalalysis implied by axioms 1 and 2 in Section 4.2.This approach derives optimal ts from the postulated individual-time points in each country.It differs from standard n, which equally weights the indicators in Section 2 by an arithmetic summation.It other indexes, which weight their indicators to maximize the prediction of external el Principal Components Analysis n treats latent individual-time points on the real line, with individuals nested within s for a given nation (Bechtel 2017;Johnston 1984, pp. 536-44).
Lemma 3. Substituting G tj for ely, from which component indicators are standardized.ximum values, each dimension index is calculated as the ratio ximum value less minimum value.ratio is first applied to each of the two indicators, and then the indices is taken.Because each dimension index is a proxy for nsion, the transformation function from income to capabilities dditional dollar of income has a smaller effect on expanding ural logarithm of the actual, minimum, and maximum values t population distributions akin to those in Coombs' theory of 4).Each of the three GDP indicators in Section 2 is posited to er individual-time points in a particular nation.Internally tructed for China and Pakistan from a latent 2-level principaloms 1 and 2 in Section 4.2.This approach derives optimal individual-time points in each country.It differs from standard hts the indicators in Section 2 by an arithmetic summation.It weight their indicators to maximize the prediction of external Analysis al-time points on the real line, with individuals nested within chtel 2017; Johnston 1984, pp. 536-44).
indicator Gtj (j = 1, 2, 3) in year t is the mean G t.j of Nt dollar = 1,…, Nt, where Nt is population size in t = 1990,…, 2015.t.j (from axiom 1) into lemma 2, the yearly GDP index is G t = and "aspirational targets," respectively, from which component indicators are standardized.Having defined the minimum and maximum values, each dimension index is calculated as the ratio of actual value less minimum value to maximum value less minimum value.
For the education dimension, this ratio is first applied to each of the two indicators, and then the arithmetic mean of the two resulting indices is taken.Because each dimension index is a proxy for capabilities in the corresponding dimension, the transformation function from income to capabilities is likely to be concave-that is, each additional dollar of income has a smaller effect on expanding capabilities.Thus, for income, the natural logarithm of the actual, minimum, and maximum values is used.

Indicator Weighting
The theory here postulates latent population distributions akin to those in Coombs' theory of psychological data (Coombs 1950(Coombs , 1964)).Each of the three GDP indicators in Section 2 is posited to take an unobservable distribution over individual-time points in a particular nation.Internally consistent GDP indexes are then constructed for China and Pakistan from a latent 2-level principalcomponents analysis implied by axioms 1 and 2 in Section 4.2.This approach derives optimal indicator weights from the postulated individual-time points in each country.It differs from standard GDP calculation, which equally weights the indicators in Section 2 by an arithmetic summation.It also differs from other indexes, which weight their indicators to maximize the prediction of external criteria.

Latent 2-Level Principal Components Analysis
This section treats latent individual-time points on the real line, with individuals nested within successive years for a given nation (Bechtel 2017;Johnston 1984, pp. 536-44).
Corollary 3.An N t -weighted principal components analysis of the dollar-denominated matrix (G tj ) has first principal component G = ows: alposts) are set in order to transform the indicators scale of 0 to 1.These goalposts act as the "natural zeros" from which component indicators are standardized.
values, each dimension index is calculated as the ratio value less minimum value.is first applied to each of the two indicators, and then the s is taken.Because each dimension index is a proxy for , the transformation function from income to capabilities onal dollar of income has a smaller effect on expanding garithm of the actual, minimum, and maximum values lation distributions akin to those in Coombs' theory of ch of the three GDP indicators in Section 2 is posited to dividual-time points in a particular nation.Internally d for China and Pakistan from a latent 2-level principaland 2 in Section 4.2.This approach derives optimal dual-time points in each country.It differs from standard e indicators in Section 2 by an arithmetic summation.It t their indicators to maximize the prediction of external sis e points on the real line, with individuals nested within 2017; Johnston 1984, pp. 536-44).
or Gtj (j = 1, 2, 3) in year t is the mean G t.j of Nt dollar , Nt, where Nt is population size in t = 1990,…, 2015.
Note that the ∑tNt values in the latent vector a composite index using their geometric mean.lized [0, 1] scale for health and education (in years) and standard of living (in llar-units) is obtained as follows: and maximum values (goalposts) are set in order to transform the indicators erent units into indices on a scale of 0 to 1.These goalposts act as the "natural zeros" al targets," respectively, from which component indicators are standardized.the minimum and maximum values, each dimension index is calculated as the ratio s minimum value to maximum value less minimum value.cation dimension, this ratio is first applied to each of the two indicators, and then the of the two resulting indices is taken.Because each dimension index is a proxy for e corresponding dimension, the transformation function from income to capabilities ncave-that is, each additional dollar of income has a smaller effect on expanding s, for income, the natural logarithm of the actual, minimum, and maximum values ta Theory ighting here postulates latent population distributions akin to those in Coombs' theory of ta (Coombs 1950(Coombs , 1964)).Each of the three GDP indicators in Section 2 is posited to rvable distribution over individual-time points in a particular nation.Internally indexes are then constructed for China and Pakistan from a latent 2-level principallysis implied by axioms 1 and 2 in Section 4.2.This approach derives optimal s from the postulated individual-time points in each country.It differs from standard , which equally weights the indicators in Section 2 by an arithmetic summation.It other indexes, which weight their indicators to maximize the prediction of external l Principal Components Analysis treats latent individual-time points on the real line, with individuals nested within for a given nation (Bechtel 2017;Johnston 1984, pp. 536-44).
Finally, the following lemma reveals the relationship between our yearly GDP index G t and other linear combinations of the Keynesian indicators G t1 , G t2 , and G t3 : Lemma 4. If N t -weighted indicators G t1 , G t2 , and G t3 in Section 2 were perfectly correlated over time, then all their linear combinations would be perfectly correlated over time: In particular, G and N t -weighted GDP t = G t1 + G t2 + G t3 would be perfectly correlated over time.
Lemma 4 provides insights into our findings in Sections 5.3 and 5.4.Its implications are developed further in Section 6.
In economics, consumer demand and producer supply for a specific product are analogues of H , price is an analogue of G, and β is an analogue of product-specific price elasticity (Johnston 1984, Appendix A-2;Samuelson and Nordhaus 1985, pp. 379-86).In psychophysics, a value of G in dollars is analogous to physical stimulus intensity, the value taken by H is analogous to sensory intensity, and β is a modality-specific stimulus effect (Luce 1959a, pp. 86-87;1959b, pp. 42-44;Luce and Galanter 1963, pp. 273-83, 291-95).The power function in axiom 3 is negatively accelerated for β < 1, which accords with the well-known diminishing marginal utility of money (Samuelson and Nordhaus 1985, pp. 411-14).
Corollary 4 enables the computation of country-specific elasticity β of H with respect to G.

Internal Consistency and Country Specificity of G
The manifest component G in corollary 3 has maximum variance among all linear combinations of population-weighted GDP indicators whose squared coefficients sum to one.This conditional maximum variance equals the first eigenvalue of the population-weighted covariance matrix of the three indicators in Section 2. This first eigenvalue, divided by the sum of the three eigenvalues of this covariance matrix, gives the proportion of variance in all three GDP indicators due to G (Johnston 1984, pp. 536-38).A second measure of internal consistency is given by the Kaiser-Meyer-Olkin Measure of Sampling Adequacy (MSA).MSA also measures the proportion of common variance among the three classic GDP indicators in Section 2. It is computed here as a population parameter because no sampling, and hence no significance testing, is done.
The eigenvalues and eigenvectors of Chinese and Pakistani covariance matrices are exhibited in Tables 1 and 2. They are produced from the first Stata command (Stata Corp. 2011) in the Appendix A. The second Stata command computes the MSA parameter, and the third command gives G.
China.The second line in Table 1 shows that principal component G accounts for 99.86% of the variance in the three GDP indicators in Section 2. China's MSA is 1.00.These two measures demonstrate that the three classic indicators possess almost perfect internal consistency in measuring the Keynesian construct for the Chinese economy.
The eigenvector in the second line of Table 2 contains the optimal national weights for GDP's three indicators in China (cf.Section 4.1).Table 2 shows that gross domestic savings most heavily weights the Chinese G.
Pakistan.A principal-components analysis of the 26 × 4 Pakistani spreadsheet shows that 99.82% of the variance in its GDP indicators is attributable to G. Pakistan's MSA is also 1.00.Again, these three classic indicators give almost perfect internal consistency for the construct G in Pakistan.However, the eigenvector in the third line of Table 2 shows a very different profile for these indicators in Pakistan than in China.Pakistani national weights reveal that G is primarily driven by household consumption, with gross domestic savings having a near zero weight in G (cf. Section 4.1).
Tables 1 and 2 demonstrate the country-specificity of G's profile, which is given by latent 2-level principal-components analysis.

H as Isoelastic G
Corollary 4 predicts the regression of lnH on lnG to be perfectly linear with slope β.This slope is the percent change in H produced by a one percent change in G (Johnston 1984, Appendix 2).Table 3 shows that this elasticity is 0.1088% in China and 0.1553% in Pakistan.The R 2 s in each country demonstrate near perfect linearity of lnH on lnG as predicted by corollary 4.This cross-national linearity suggests the use of isoelastic G as a measure of H.  4 shows that near perfect indicator correlations in China produce a correlation of 1.0000 between G and N t -weighted GDP t .The slightly lower Pakistani correlation 0.9997 is due to the somewhat lower indicator correlations in Pakistan.
The two correlations in the last line of Table 4 support the use of additive GDP in computing the elasticity of a nation's HDI with respect to its gross domestic product.This substitution of N t -weighted GDP t for principal component G in axiom 3 is confirmed in Section 5.4.0.9977 0.8367 r(G, GDP) = 1.0000 r(G, GDP) = 0.9997

Isoelasticity of H and N t -Weighted GDP t
As is expected from Table 4, Table 5 confirms that N t -weighted GDP t returns almost identical elasticities and R 2 s as those given by G in Table 3.The elasticities in Table 5, like those in Table 3, demonstrate a sharply diminishing marginal utility for money in both Chinese and Pakistani societies.These tables support axiom 3, which posits H as an isoelastic power function of G.There is little change in H beyond its sizable increases driven by initial increments in G.This phenomenon is graphed in Figures 1 and 2. The more precise Chinese function in Figure 1 is due to the higher indicator correlations for China in Table 4.

Conclusions
The present paper discovers three new properties of the gross domestic product in very different Asian economies: First, Table 4  Second, this principal component G is in a class of weighted indexes ancillary to GDP.Each ancillary index informs policy as to allocation of resources over the three GDP indicators.The differential weighting of indexes like Gt is country specific, as is shown in Table 2 for China and

Conclusions
The present paper discovers three new properties of the gross domestic product in very different Asian economies: First, Table 4 exhibits very high correlations over time among GDP indicators in China and Pakistan.Lemma 4 shows that this level of internal consistency implies near perfect correlations among all linear combinations of these indicators.Table 4 confirms, in particular, that Keynesian additive GDP is almost perfectly correlated over time with the first principal component G of the GDP indicators.
Second, this principal component G is in a class of weighted indexes ancillary to GDP.Each ancillary index informs policy as to allocation of resources over the three GDP indicators.The differential weighting of indexes like Gt is country specific, as is shown in Table 2 for China and

Conclusions
The present paper discovers three new properties of the gross domestic product in very different Asian economies: First, Table 4 exhibits very high correlations over time among GDP indicators in China and Pakistan.Lemma 4 shows that this level of internal consistency implies near perfect correlations among all linear combinations of these indicators.Table 4 confirms, in particular, that Keynesian additive GDP is almost perfectly correlated over time with the first principal component G of the GDP indicators.
Second, this principal component G is in a class of weighted indexes ancillary to GDP.Each ancillary index informs policy as to allocation of resources over the three GDP indicators.The differential weighting of indexes like G t is country specific, as is shown in Table 2 for China and Pakistan.The International Monetary Fund (IMF), especially sensitive to China's global effects, noted that Chinese government policy is now "designated to accelerate the transformation of the Chinese economic model, improve livelihoods, and raise domestic consumption" (Singh et al. 2013).The generalized principal components analysis in Table 2 confirms the needed reallocation of Chinese GDP from gross domestic savings to household consumption.This finding illustrates how weighted indexes like G t = a 1 G t1 + a 2 G t2 + a 3 G t3 can inform governments about their GDP allocation.It also raises new questions as to which weighting of G t1 , G t2 , and G t3 is most informative to policy.The latent 2-level principal components analysis in Section 4.2 selects G t = a 1 G t1 + a 2 G t2 + a 3 G t3 in Lemma 3, where a 1 , a 2 , and a 3 are exhibited in Table 2 for China and Pakistan.This optimal internal weighting (a 1 , a 2 , a 3 ) differs from that of an index that weights G t1 , G t2 , and G t3 to maximize the prediction of external criteria.Nonetheless, alternative policy guidance is given by an external index that also has near perfect time-series correlation with unweighted GDP t = G t1 + G t2 + G t3 .
Third, a country-specific power of GDP almost perfectly predicts HDI.This finding is brought by regression analyses of time-series supplied by the World Bank and the United Nations.It is concluded that axiomatic HDI computation can be carried out in China and Pakistan without survey sampling, probabilistic inference, significance testing, or even HDI data.
Finally, the open problem of sustained GDP growth has been studied by the Leeds UK Steady State Economy Conference (O'Neill et al. 2010), the United Nations Division for Sustainable Development (Costanza et al. 2012), and the Annual Forum of The Progressive Economy Initiative (Journal for a Progressive Economy 2015).This problem has also been addressed by the Sarkozy report, which ends with the admonition that "no limited set of figures can pretend to forecast the sustainable or unsustainable character of a highly complex system" (Stiglitz et al. 2010, p. 136).Alperovitz envisions the present "political, ecological, and economic" system to be "the prehistory of transformative and fundamental systemic change. . . .Sustainability requires . . .a transformative vision beyond both corporate capitalism and traditional state socialism."(Alperovitz 2017) (Italics mine).
t.j of N t dollar denominated impacts 3 of 10 goods and services (including compensation of es on national defense and security, but excludes of government capital formation.Data are in current are described by the United Nations Development erage achievement in key dimensions of human wledgeable, and having a decent standard of living.ndices for each of the three dimensions.e expectancy at birth, the education dimension is ults aged 25 years and more and expected years of e standard of living dimension is measured by gross e logarithm of income to reflect the diminishing scores for the three HDI dimension indices are then eometric mean.d education (in years) and standard of living (in s: tion's GDP indicator Gtj (j = 1, 2, 3) in year t is the mean G t.j of Nt dollar individuals i = 1,…, Nt, where Nt is population size in t = 1990,…, 2015.ti1 eory g postulates latent population distributions akin to those in Coombs' theory of oombs 1950, 1964).Each of the three GDP indicators in Section 2 is posited to le distribution over individual-time points in a particular nation.Internally xes are then constructed for China and Pakistan from a latent 2-level principalimplied by axioms 1 and 2 in Section 4.2.This approach derives optimal m the postulated individual-time points in each country.It differs from standard ich equally weights the indicators in Section 2 by an arithmetic summation.It r indexes, which weight their indicators to maximize the prediction of external ncipal Components Analysis ts latent individual-time points on the real line, with individuals nested within

=
of the actual, minimum, and maximum values istributions akin to those in Coombs' theory of three GDP indicators in Section 2 is posited to -time points in a particular nation.Internally ina and Pakistan from a latent 2-level principalin Section 4.2.This approach derives optimal e points in each country.It differs from standard ors in Section 2 by an arithmetic summation.It ndicators to maximize the prediction of external on the real line, with individuals nested within nston 1984, pp.536-44).= 1, 2, 3) in year t is the mean G t.j of Nt dollar re Nt is population size int = 1990,…, 2015.b + rithm of the actual, minimum, and maximum values ion distributions akin to those in Coombs' theory of of the three GDP indicators in Section 2 is posited to idual-time points in a particular nation.Internally r China and Pakistan from a latent 2-level principald 2 in Section 4.2.This approach derives optimal al-time points in each country.It differs from standard dicators in Section 2 by an arithmetic summation.It heir indicators to maximize the prediction of external oints on the real line, with individuals nested within 7;Johnston 1984, pp.536-44).Gtj (j = 1, 2, 3) in year t is the mean G t.j of Nt dollar t, where Nt is population size int = 1990,…, 2015.w , where tural logarithm of the actual, minimum, and maximum values nt population distributions akin to those in Coombs' theory of 64).Each of the three GDP indicators in Section 2 is posited to over individual-time points in a particular nation.Internally structed for China and Pakistan from a latent 2-level principalioms 1 and 2 in Section 4.2.This approach derives optimal individual-time points in each country.It differs from standard ghts the indicators in Section 2 by an arithmetic summation.It h weight their indicators to maximize the prediction of external ts Analysis ual-time points on the real line, with individuals nested within echtel 2017;Johnston 1984, pp.536-44).

Figure 1 .
Figure 1.Human Development Index (HDI) is in [0, 1] and GDP is in billions of dollars.

Figure 2 .
Figure 2. HDI is in [0, 1] and GDP is in billions of dollars.

Figure 1 .
Figure 1.Human Development Index (HDI) is in [0, 1] and GDP is in billions of dollars.

Figure 2 .
Figure 2. HDI is in [0, 1] and GDP is in billions of dollars.
HDI is in [0,1] and GDP is in billions of dollars.HDI is in [0,1] and GDP is in billions of dollars.Pakistani HDI as a Function of GDP

Figure 2 .
Figure 2. HDI is in [0, 1] and GDP is in billions of dollars.
It also includes most expenditures on national defense and security, but excludes government military expenditures that are part of government capital formation.Data are in current U.S. dollars.
in year t is the mean G t.j of Nt dollar , where Nt is population size int = 1990,…, 2015.tij on individuals i = 1, . . .
) in year t is the mean G t.j of Nt dollar denominated impacts Gtij on individuals i = 1,…, Nt, where Nt is population size int = 1990,…, 2015.
year t is the mean G t.j of Nt dollar cts Gtij on individuals i = 1,…, Nt, where Nt is population size int = 1990,…, 2015.
year t is the mean G t.j of Nt dollar denominated impacts Gtij on individuals i = 1,…, Nt, where Nt is population size int = 1990,…, 2015.

Table 1 .
Eigenvalues for G.Note.The proportions of indicator variance accounted for by the first principal components are 0.9986 in China and 0.9982 in Pakistan.Each proportion equals the first eigenvalue divided by the sum of that nation's three eigenvalues.

Table 2 .
First eigenvectors for G.

Table 3 .
Regressions of lnH on lnG.Lemma 4 implies that the internal consistency of N t -weighted indicators G t1 , G t2 , and G t3 governs the correlation between G and N t -weighted GDP t

Table 4 .
Correlations of N t -weighted indicators, G, and N t -weighted GDP t .

Table 5 .
Regressions of lnH on ln{N t -weighted GDP t }.
exhibits very high correlations over time among GDP indicators in China and Pakistan.Lemma 4 shows that this level of internal consistency implies near perfect correlations among all linear combinations of these indicators.Table 4 confirms, in particular, that Keynesian additive GDP is almost perfectly correlated over time with the first principal component G of the GDP indicators.