I. Introduction

Not many papers on the influence of demographic variables on consumers’ behavior have generated as much controversy as the paper by Leff (1969) on the inverse relationship between dependency rates and savings rates. As it is well known, this conclusion follows from the observation that a rapidly growing population has a large number of young people, who tend to consume more than they produce. In the absence of an offsetting increase in the income of adults or decrease in the consumption of adults, the effect will be to reduce aggregate savings. Resources available for investment would therefore be reduced by the added burden of dependents. In Leff’s paper, dependency ratios (i.e., the proportions of the population under 15 and over 65 years of age) are used, along with per capita income and the growth rate of income, as explanatory variables and are found to decrease the savings rate significantly in a cross-section of 74 countries, of which 47 are developing countries.

Commenting on Leff’s paper, a number of authors (Adams (1971), Gupta (1971), Goldberger (1973), and Bilsborrow (1979, 1980)) have pointed out that Leff’s analysis is likely to be affected by serious shortcomings mainly originating from omitted variables, sample selection, and, in general, the nature and quality of the data. Replying to his critics, Leff (1971, 1973, 1980) has strongly argued in favor of its original results, suggesting that the conclusions of his paper still stand.

Leff and his critics agree, however, on one thing: the existing evidence lacks a fully specified theoretical framework. ^2/ In addition, sample heterogeneity, the widespread use of a single cross-section of possibly weak data, as well as the inappropriate treatment of some econometric issues, all imply that upon careful re-examination, the available evidence cannot be taken for granted.

Some of these criticisms also apply to the recent paper by Ram (1982). As Leff (1984) has also pointed out, ^3/ in Ram’s work as in all previous work, a theoretical framework is not clearly spelled out in detail so that we simply do not know, for example, whether or not his regression coefficients for the variables of interest are plausible or, indeed, why the relationship between dependency rates and savings rates shows up in some groups of countries and not in others.

The purpose of this paper is to reconsider the question of the relationship between dependency rates and savings in the context of a properly defined theoretical framework applicable to household’s savings behavior under uncertainty. Two demographic extensions of the standard representative household’s stochastic dynamic optimization problem are formulated. Both are derived from the demand analysis literature and correspond to a restricted version of well-known procedures for introducing demographic variables into demand systems. The first model simply expresses intertemporal preferences in terms of consumption per "equivalent" adult. In the second model, preferences are, instead, expressed in terms of per capita consumption net of demographically varying overhead cost. ^4/

The Euler equations corresponding to the two representations of the representative household’s preferences are derived in Section II. It is shown that, while both models point to a relationship between the expected dependency rate and the growth rate of consumption, they have substantially different implications as far as their magnitude and direction are concerned. In particular, the first and more commonly used model (i.e., the equivalent adult model) only allows a negative relationship to exist whose magnitude is determined by the so-called "cost-of-children" (Deaton and Muellbauer (1985)), that is, by the scale that seeks to quantify and represent in one summary measure the changing needs of a family as it expands and changes its composition. In the second (i.e., the overhead cost model), instead, the relationship between expected changes in dependency rates and consumption is controlled by two parameters: the intertemporal elasticity of substitution and a measure of the share of committed consumption, so that, since the latter is positive but smaller than unity, expected changes in dependency rates can have positive, negative, or no effect whatsoever on the growth rate of consumption depending on the consumers’ willingness to keep a smooth profile of consumption even in the face of expected changes in family composition. Notice that, in the latter model, if independent information exists on the share of committed consumption, the relationship between dependency rates and the growth rate of consumption provides information on the intertemporal elasticity of substitution. In developing countries, where the size of the capital market is small and usually confined to one central city, and wealth is held in the form of consumer durables such as jewelry and livestock, rates of return on financial instruments are likely to be largely irrelevant and rates of return on physical assets are likely to be unobservable. In such a case, information on future family composition can be of substantial help.

Empirical evidence is provided in Section III, based on data on private savings over the period 1973-83 in 49 developing countries, that is, where the issue has the greatest relevance. Data are grouped in six sets of pooled time-series cross-section observations, each one referring to a single geographical region. It is hoped that the pooling of cross-section and time-series data can help overcome the sample heterogeneity problem faced by all the previous works on the subject and pointed out by Bilsborrow (1980, pp. 186-89). Finally, Section IV contains a few concluding remarks.

II. Theory

The most recent research on consumption (reviewed in Deaton (1985), and King (1985)) has been marked by a number of important works (Hall (1978), Grossman and Shiller (1981), and Hansen and Singleton (1982)) opening the possibility of a direct estimation of the parameters of the intertemporal utility function characterizing the behavior of an (infinitely-lived) representative consuming unit without requiring explicit solutions to the dynamic optimization problem. ^5/

Following this line of research, we posit that aggregate consumption can be modeled as the outcome of optimizing decisions of a representative consumption unit (household). The household faces an economic environment where future opportunities are uncertain, and its stationary utility function is given by:

In equation (1), V_t is expected utility at t, E_t is the expectations operator conditional on information available at t, ρ is a constant discount factor, and F_t is actual family size. The parameter γ in equation (1) controls intertemporal substitution: large and negative values of γ characterize consumers who are willing to smooth consumption over time and who respond only to substantial changes in incentives. Finally, the function U_τ(.) takes, alternatively, one of the two following forms:

or

In equation (2), U_τ denotes household’s consumption in terms of equivalent adults since E_τ is "effective" household size defined as follows:

where d_τ is the proportion of household members in the young age bracket (for example, 0-14) or, in other words, the household’s dependency rate. ^6/ In addition, λ is a proportionality factor converting children to equivalent adults for consumption purposes. ^7/

Equation (2) has been used by a number of authors to describe the effect of changing family size on private consumption (Modigliani and Ando (1957) and, more recently, Mariger (1986)). It implies that the within-period utility function is concave and increasing in private consumption per equivalent household member and it corresponds to defining intertemporal utility in terms of scaled quantities (C_τ/E_τ). Notice that the λ parameter (otherwise called "the cost of children" or "general equivalence scale") is independent of time.

In equation (3), instead, demographic variables enter household’s preferences through the committed quantity or overhead cost ${\overline{\mathrm{c}}}_{\mathrm{\tau }}$ which is assumed to be linearly dependent on the number of young household members (Y_τ) as follows:

Clearly, equation (5) implies an intertemporal utility function expressed in terms of uncommitted quantities $\mathrm{\left(}{\mathrm{C}}_{\mathrm{\tau }}\mathrm{-}{\overline{\mathrm{c}}}_{\mathrm{\tau }\mathrm{,}}\mathrm{\forall }\mathrm{\tau }\mathrm{\right)}$.

The household expects, for simplicity, the real rate of return to remain constant over time, and maximizes equations (1), (2), and (4) (or (1), (3), and (5)) subject to the usual intertemporal budget constraints equating the discounted present value of consumption to the discounted present value of assets and non-interest income or, in period-to-period form:

where A_τ defines real assets at the end of period τ, R is the constant real rate of return, and Y_τ is real net nonproperty income in period τ. It is assumed that some asset exists which is either held in positive amounts or for which borrowing is possible. As long as the optimum path lies in the interior of the budget set, we can use simple perturbation arguments to establish certain characteristics of the optimal path. At any point along an optimal path the representative consumption unit cannot make itself better-off by forgoing one unit of consumption at time t and using the proceeds to purchase any other good at any other point in time. Formally, at time t the marginal condition will be given by:

which, apart from implicitly defining F_t+1, is satisfied for any free traded risky asset and holds for consumers who expect with certainty to be there in the next period.

Under rational expectations and market clearing, the first-order condition (9) holds ex post except for an error term uncorrelated with information available to the consumption unit at time t. In other words:

where ε_t+1 is the mean zero and constant variance (σ²) forecast error.

In the present case of the time separable, constant relative risk averse preferences, and letting lower-case letters denote natural logarithms and Δ be the difference operator, both specifications of the within-period utility function imply: ^8/

in terms of consumption per (actual) household member, and where α = α(ρ, σ², γ, r), and

ϕ = (λ - 1)

if (2) and (4) hold

View Table

ϕ = (λ - 1)

if (2) and (4) hold

while

ϕ = -γ ω/(1-γ)

if (3) and (5) hold

View Table

ϕ = -γ ω/(1-γ)

if (3) and (5) hold

Therefore, in the equivalent adult model, -1 < ϕ < 0 and, in the light of the available evidence (Deaton and Muellbauer (1986)), could well be around -.6. In the case of the overhead cost model, instead, for ω = 1, ϕ would be equal to the coefficient of the expected real interest rate minus one if the latter would not have been assumed constant. In other words, for a given value of ω, ϕ provides direct information on the consumers’ willingness to smooth their consumption path and, therefore, other things being equal, on the real interest rate elasticity of consumption. Since ω = [(β_τ F_τ)/C_τ], it can be regarded as the share of committed consumption (evaluated at the needs of the young members of the household) over total consumption. In what follows, we shall assume ω to be independent of τ. Notice that ϕ is greater, equal, or less than zero depending on whether γ is negative, zero, or positive, respectively. In addition, if the household strongly prefers an even consumption path (i.e., γ → -∞) the extent of the relationship between consumption growth and expected dependency rate is given by the relative amount of overhead costs (i.e., ϕ → ω) and the rate of growth of "net" consumption is a constant. On the other hand, if the household tends to be extremely responsive to changes in the incentives (i.e., γ → 1), consumption is likely to be shifted backward toward the present period, if effective expected family size is expected to increase (i.e., ϕ → -∞). Ceteris paribus, this will imply lower savings in the current period t. Therefore, the magnitude of the coefficient ϕ turns out to possess useful information on the consumers’ willingness to smooth consumption over time.

In equation (11), the error term u_t+1 reflects the impact of "news" about current levels of income, and household size and composition. It is therefore orthogonal to all past information. ^9/

As it stands, equation (11) refers to the case of a consuming unit executing intertemporal optimization through trading in perfectly competitive asset markets. The available empirical evidence (Rossi (1987)) strongly suggests, however, that a significant fraction of the population in developing countries is not generally able to shift consumption at will from a later date to an earlier date, as assumed by the well-known life-cycle theory for a number of reasons, among which capital market imperfections. While the gravity of capital market imperfections continues to be a matter of debate even in countries with apparently sophisticated financial institutions and well-developed capital markets, there are several reasons why such imperfections are likely to be exacerbated in developing countries. Following Muellbauer (1986) and the empirical analysis in Rossi (1987), allowance is therefore made for departure from optimal behavioral rules by the representative consuming unit described by the theory, by simply adding a term [+ ζ E_t (z_t+1-c_t)] to equation (11), where Z_t is real net nonproperty income. Consumers’ liquidity constrained at t may not expect to be constrained at t+1 and may therefore be forced to let their consumption path follow more closely their income path. The coefficient ζ can be expected to be positive and significantly different from zero if liquidity constraints are present (see Muellbauer (1986) for additional details).

III. Data and Estimates

A thorough empirical analysis of private savings behavior in developing countries raises several difficult statistical problems, mostly stemming from the inadequacies in the data and their lack of comparability. A reasonable number of observations on aggregate time-series data is available, on a consistent basis, only for a few developing countries. In the great majority of cases only less than 20 annual observations are available. In such a situation, pooling cross-section and time-series data for a number of countries seems to be the most sensible procedure if, however, allowance is made as much as possible for obvious institutional and cultural differences among countries.

Following this line of research, the empirical analysis of the present paper is based on six sets of pooled time-series cross-section data, each one referring to a geographical region, hopefully homogeneous. In particular, the first set covers 12 countries in Sub-Saharan Africa, the second set includes 5 countries in North Africa and the Middle East, and the third one covers 9 countries in East and South Asia and the Pacific. The fourth and fifth sets cover 8 and 9 countries in Central America (including the Caribbean) and South America, respectively. Finally, the sixth includes 6 southern European countries. It should be noticed that the sample as a whole contains 11 low-income and 38 middle-income countries. Low-income countries are, therefore, somewhat underrepresented. ^10/

The Data Appendix in Rossi (1987) provides a description of the data set which has been constructed by assembling information from all available international, as well as national, sources as needed.

For estimation purposes, let us rewrite the model derived in Section II as follows:

where the suffix "i" identifies the i-th country in each of the geographical areas referred to in the previous section. In other words, the constant term in equation (11), being a function of the variance of the forecast error, is allowed to differ among countries because, for example, countries with a higher share of the product originating from agriculture are likely to face higher uncertainty. In addition, the original error term in equation (11), i.e., ${\mathrm{u}}_{\mathrm{t}\mathrm{+}\mathrm{1}}^{\mathrm{1}}\mathrm{,}$ is now linearly decomposed in two random components with mean zero but not necessarily homoskedastic, because the variances of different country forecast errors are likely to differ. The first component is country-specific and is uncorrelated across countries, $\mathrm{\left(}{\mathrm{v}}_{\mathrm{t}\mathrm{+}\mathrm{1}}^{\mathrm{1}}\mathrm{\right)}\mathrm{,}$ while the second one is an area-wide component which equally affects all countries in a particular geographical area $\mathrm{\left(}{\overline{\mathrm{v}}}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{\right)}\mathrm{.}$. ^11/ The obvious example of the latter component is given by the recent drought in Sub-Saharan Africa.

Disregarding, for the time being, the expected nature of the variables on the right-hand side of equation (11), the appropriate estimator for the kind of setting described by equation (11) is given by what is known as the between-within groups fixed-effects estimator, if we can regard each country as a group. As Mundlak (1978) shows, this estimator amounts to apply Ordinary Least Squares to equation (12) expressed in terms of "transformed" variables, where the transformation takes the following form:

for a generic variable ${\mathrm{x}}_{\mathrm{t}}^{\mathrm{i}}\mathrm{,}$ and if T and N denote the number of time periods and the number of countries, respectively. That is, the transformed variable is the original variable minus the country and time means plus the total mean. Notice that the transformation eliminates the constant term and the area-wide term. In general, the transformation would eliminate all variables not simultaneously indexed on i and t.

Reverting now to equation (12), its coefficients can be estimated, after transforming the variables, using instrumental variables for Δd_t+1 and z_t+1. All variables known at time t are good candidates as instruments. Table 1 reports the parameter estimates for the six sub-samples, their White’s (1980) heteroskedasticity consistent standard errors (in parentheses), the standard error of the regression (σ), as well as some diagnostic statistics like the Durbin Watson statistics for fixed-effects model given in Bhargava and others (1982), denoted by "dw", and the test of the over-identifying restrictions (including the validity of instruments) due to Sargan (1984) (X²(.)). ^12/ The set of instruments includes lagged consumption, lagged real net nonproperty income, dependency rate lagged once and twice, total population lagged once and twice, and a time trend. In addition, assuming the overhead cost model to be correct, the table reports the values of γ that would be implied by the estimated ϕ’s if ω could be alternatively given the value 0.3 or 0.7. For each geographical region, two sets of estimates are presented. The first one (rows denoted by (i)) refers to equation (12) above, but with ζ set to zero (no liquidity constraints). The second one, instead, (rows (ii)) allows for liquidity constraints.

Table 1.

Parameter Estimates and Test Statistics

^1/The regressions also include a dummy variable taking a value of 1 in 1973 and of -1 in 1984 for Swaziland. It accounts for two large outliers but does not affect the remaining coefficients. Its coefficient takes a value of -0.51 (0.02) and -0.41 (0.08) in the regressions of rows (i) and (ii), respectively.

^2/With 6 degrees of freedom.

^3/With 5 degrees of freedom.

^4/The regressions also include two dummy variables taking a value of 1 in 1974 for both Cyprus and Portugal. They do not affect the remaining coefficients and their coefficients take the following values: -0.20 (0.01), -0.10 (0.02) for the first dummy variable in the regressions of rows (i) and (ii), respectively; 0.09 (0.02), 0.08 (0.01) for the second dummy variable in the regressions of rows (i) and (ii), respectively.

Table 1.

Parameter Estimates and Test Statistics

			ϕ	ζ	σ	dw	x2(.)	$\frac{\mathrm{\gamma }}{\mathrm{\left(}\mathrm{\omega }\mathrm{=}\mathrm{0.3}\mathrm{\right)}\mathrm{\left(}\mathrm{\omega }\mathrm{=}\mathrm{0.7}\mathrm{\right)}}$
Africa South of Sahara
	(n.ob.: 104) 1/
		(i)	-2.41 (3.51)	--	0.070	1.75	20.9 2/	0.9	0.8
		(ii)	-1.70 (2.97)	0.37 (.10)	0.057	1.80	5.5 3/	0.9	0.7
Middle East and North Africa
	(n.ob.: 44)
		(i)	-0.42 (4.66)	--	0.072	1.81	25.3 2/	0.3	-1.3
		(ii)	-0.42 (4.68)	0.28 (.09)	0.061	1.66	23.8 3/	0.3	-1.3
East and South Asia and the Pacific
	(n.ob.: 84)
		(i)	-1.27 (2.36)	--	0.033	2.11	4.0 2/	0.8	0.6
		(ii)	-1.63 (2.43)	0.21 (1.2)	0.026	2.14	3.5 3/	0.8	0.7
Southern Europe
	(n.ob.: 56) 4/
		(i)	-0.44 (3.56)	--	0.033	2.11	4.0 2/	0.8	-1.3
		(ii)	-4.92 (2.74)	0.21 (0.12)	0.036	2.14	3.5 3/	0.9	0.9
Central America and the Caribbean
	(n.ob.: 74)
		(i)	9.23 (5.66)	--	0.048	1.14	6.1 2/	--	--
		(ii)	9.23 (5.73)	.01 (.29)	0.048	1.15	6.1 3/	--	--
South America
	(n.ob.: 88)
		(i)	-0.38 (6.02)	--	0.053	1.56	40.2 2/	0.3	-1.3
		(ii)	-2.42 (4.33)	0.75 (0.18)	0.038	1.18	12.8 3/	0.9	0.8

^1/The regressions also include a dummy variable taking a value of 1 in 1973 and of -1 in 1984 for Swaziland. It accounts for two large outliers but does not affect the remaining coefficients. Its coefficient takes a value of -0.51 (0.02) and -0.41 (0.08) in the regressions of rows (i) and (ii), respectively.

^2/With 6 degrees of freedom.

^3/With 5 degrees of freedom.

^4/The regressions also include two dummy variables taking a value of 1 in 1974 for both Cyprus and Portugal. They do not affect the remaining coefficients and their coefficients take the following values: -0.20 (0.01), -0.10 (0.02) for the first dummy variable in the regressions of rows (i) and (ii), respectively; 0.09 (0.02), 0.08 (0.01) for the second dummy variable in the regressions of rows (i) and (ii), respectively.

View Table

Table 1.

Parameter Estimates and Test Statistics

			ϕ	ζ	σ	dw	x2(.)	$\frac{\mathrm{\gamma }}{\mathrm{\left(}\mathrm{\omega }\mathrm{=}\mathrm{0.3}\mathrm{\right)}\mathrm{\left(}\mathrm{\omega }\mathrm{=}\mathrm{0.7}\mathrm{\right)}}$
Africa South of Sahara
	(n.ob.: 104) 1/
		(i)	-2.41 (3.51)	--	0.070	1.75	20.9 2/	0.9	0.8
		(ii)	-1.70 (2.97)	0.37 (.10)	0.057	1.80	5.5 3/	0.9	0.7
Middle East and North Africa
	(n.ob.: 44)
		(i)	-0.42 (4.66)	--	0.072	1.81	25.3 2/	0.3	-1.3
		(ii)	-0.42 (4.68)	0.28 (.09)	0.061	1.66	23.8 3/	0.3	-1.3
East and South Asia and the Pacific
	(n.ob.: 84)
		(i)	-1.27 (2.36)	--	0.033	2.11	4.0 2/	0.8	0.6
		(ii)	-1.63 (2.43)	0.21 (1.2)	0.026	2.14	3.5 3/	0.8	0.7
Southern Europe
	(n.ob.: 56) 4/
		(i)	-0.44 (3.56)	--	0.033	2.11	4.0 2/	0.8	-1.3
		(ii)	-4.92 (2.74)	0.21 (0.12)	0.036	2.14	3.5 3/	0.9	0.9
Central America and the Caribbean
	(n.ob.: 74)
		(i)	9.23 (5.66)	--	0.048	1.14	6.1 2/	--	--
		(ii)	9.23 (5.73)	.01 (.29)	0.048	1.15	6.1 3/	--	--
South America
	(n.ob.: 88)
		(i)	-0.38 (6.02)	--	0.053	1.56	40.2 2/	0.3	-1.3
		(ii)	-2.42 (4.33)	0.75 (0.18)	0.038	1.18	12.8 3/	0.9	0.8

^1/The regressions also include a dummy variable taking a value of 1 in 1973 and of -1 in 1984 for Swaziland. It accounts for two large outliers but does not affect the remaining coefficients. Its coefficient takes a value of -0.51 (0.02) and -0.41 (0.08) in the regressions of rows (i) and (ii), respectively.

^2/With 6 degrees of freedom.

^3/With 5 degrees of freedom.

^4/The regressions also include two dummy variables taking a value of 1 in 1974 for both Cyprus and Portugal. They do not affect the remaining coefficients and their coefficients take the following values: -0.20 (0.01), -0.10 (0.02) for the first dummy variable in the regressions of rows (i) and (ii), respectively; 0.09 (0.02), 0.08 (0.01) for the second dummy variable in the regressions of rows (i) and (ii), respectively.

Apart from confirming the importance of the proxy for liquidity constraints, which turns out to be irrelevant only in the case of Central America and the Caribbean, ^13/ Table 1 provides a number of interesting indications. First of all, in all cases except, again, Central America and the Caribbean, -∞ < ϕ < ω, as predicted by the more general case given by equations (3) and (5), that is, by the overhead cost model. Only in one case, the point estimate of ϕ is between 0 and -1 as also the equivalent adult model would predict. ^14/ It remains true, however, that except for Southern Europe, the estimates of ϕ are highly imprecise thereby precluding any definite statement.

Interestingly, though, the implied estimates of γ suggest a very high degree of intertemporal substitution, far higher than the one estimated on the basis of the relationship between the rate of growth of consumption and the expected real rate of return on financial assets (see Rossi (1987)).

IV. Concluding Comments

This paper has presented two simple demographic extensions of the representative household’s stochastic dynamic optimization problem designed to clarify the long-debated issue of the relationship between the savings rate and the household dependency rate in developing countries. The implications of this relationship cannot be overemphasized, given the present state of international capital markets. Lower rates of domestic savings cannot call anymore for an increased reliance on foreign capital and, therefore, growth targets have to be revised downward.

The paper has shown that the magnitude of the above relationship can well depend on the consumers’ willingness to smooth consumption over time and, therefore, can depend on another debated issue: the degree of intertemporal substitution in developing countries. In turn, this can help explain why more controversies are generated by the available studies than are answered, with particular reference to the sample heterogeneity problem faced by most authors.

Therefore, the empirical evidence provided in the paper is only apparently disappointing in that, with the only exception of Southern Europe, the estimates of the coefficients linking the rate of growth of consumption to the expected change in the dependency rate are highly imprecise, thereby precluding any definite statement. As it turns out (Rossi (1987)), Southern Europe is, in fact, one of those few regions for which independent evidence has suggested that the degree of intertemporal substitution is likely to be sizable and accurately estimated.

Finally, the methodology outlined in the paper suggests that, in developing countries where rates of return on financial assets are unobservable or irrelevant, the expected path of demographic variables can provide information on the savings responsiveness to changes in the real interest rate.

Variable Definitions

ct	=	private final consumption expenditure, in constant prices.
			(Source: United Nations, National Accounts Statistics: Main Aggregates and Detailed Tables, 1983, New York, 1986
		(Tables 1.1 and 1.2).)
Ft	=	total population (mid-year estimates).
			(Source: World Bank, Economic Analysis and Projections Department, Social Indicators of Development.)
Yy	=	total population in ages 0-14.
			(Source: same as for Ft.)
Zt	=	per capita private net nonproperty income, in constant prices. Defined as gross national product less consumption of fixed capital (when available), plus net transfers from abroad (when available), less tax revenue, plus subsidies and current transfers (when available), deflated by private final consumption implicit price index.
			(Sources: United Nations, National Accounts Statistics: Main Aggregates and Detailed Tables, 1983, New York, 1986 (Table 1.12) for gross national products, consumption of fixed capital, and net current transfers from abroad; United Nations, National Accounts Statistics: Main Aggregates and Detailed Tables, 1983, New York, 1986 (Table 1.4), and/or International Monetary Fund, Government Finance Statistics Yearbook, Washington, D.C., 1985 (Summary Table and Table C) for tax revenue and subsidies and current transfers; World Bank, Economic Analysis and Projections Department, Social Indicators of Development, for population data.)

View Table

ct	=	private final consumption expenditure, in constant prices.
			(Source: United Nations, National Accounts Statistics: Main Aggregates and Detailed Tables, 1983, New York, 1986
		(Tables 1.1 and 1.2).)
Ft	=	total population (mid-year estimates).
			(Source: World Bank, Economic Analysis and Projections Department, Social Indicators of Development.)
Yy	=	total population in ages 0-14.
			(Source: same as for Ft.)
Zt	=	per capita private net nonproperty income, in constant prices. Defined as gross national product less consumption of fixed capital (when available), plus net transfers from abroad (when available), less tax revenue, plus subsidies and current transfers (when available), deflated by private final consumption implicit price index.
			(Sources: United Nations, National Accounts Statistics: Main Aggregates and Detailed Tables, 1983, New York, 1986 (Table 1.12) for gross national products, consumption of fixed capital, and net current transfers from abroad; United Nations, National Accounts Statistics: Main Aggregates and Detailed Tables, 1983, New York, 1986 (Table 1.4), and/or International Monetary Fund, Government Finance Statistics Yearbook, Washington, D.C., 1985 (Summary Table and Table C) for tax revenue and subsidies and current transfers; World Bank, Economic Analysis and Projections Department, Social Indicators of Development, for population data.)

All mentioned information is available on diskettes to be used with Lotus 1-2-3, a "spreadsheet" program for the IBM-PC. See Rossi (1987, Appendix I) for further details.